Displaying 1-10 of 21 results found.
Triangle read by rows in which row n lists the widths of the symmetric representation of sigma(n).
+10
63
1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1
COMMENTS
Here T(n,k) is defined to be the "k-th width" of the symmetric representation of sigma(n), with n>=1 and 1<=k<=2n-1. Explanation: consider the diagram of the symmetric representation of sigma(n) described in A236104, A237593 and other related sequences. Imagine that the diagram for sigma(n) contains 2n-1 equidistant segments which are parallel to the main diagonal [(0,0),(n,n)] of the quadrant. The segments are located on the diagonal of the cells. The distance between two parallel segment is equal to sqrt(2)/2. T(n,k) is the length of the k-th segment divided by sqrt(2). Note that the triangle contains nonnegative terms because for some n the value of some widths is equal to zero. For an illustration of some widths see Hartmut F. W. Hoft's contribution in the Links section of A237270. Row n has length 2*n-1.
If n is a power of 2 then all terms of row n are 1's.
If n is an even perfect number then all terms of row n are 1's except the middle term which is 2.
If n is an odd prime then row n lists (n+1)/2 1's, n-2 zeros, (n+1)/2 1's.
The number of blocks of positive terms in row n gives A237271(n). The sum of the k-th block of positive terms in row n gives A237270(n,k). It appears that the trapezoidal numbers ( A165513) are also the numbers k > 1 with the property that some of the noncentral widths of the symmetric representation of sigma(k) are not equal to 1. - Omar E. Pol, Mar 04 2023
EXAMPLE
Triangle begins:
1;
1,1,1;
1,1,0,1,1;
1,1,1,1,1,1,1;
1,1,1,0,0,0,1,1,1;
1,1,1,1,1,2,1,1,1,1,1;
1,1,1,1,0,0,0,0,0,1,1,1,1;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1;
1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1;
1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1;
1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1;
...
---------------------------------------------------------------------------
. Written as an isosceles triangle Diagram of
. the sequence begins: the symmetry of sigma
---------------------------------------------------------------------------
. _ _ _ _ _ _ _ _ _ _ _ _
. 1; |_| | | | | | | | | | | |
. 1,1,1; |_ _|_| | | | | | | | | |
. 1,1,0,1,1; |_ _| _|_| | | | | | | |
. 1,1,1,1,1,1,1; |_ _ _| _|_| | | | | |
. 1,1,1,0,0,0,1,1,1; |_ _ _| _| _ _|_| | | |
. 1,1,1,1,1,2,1,1,1,1,1; |_ _ _ _| _| | _ _|_| |
. 1,1,1,1,0,0,0,0,0,1,1,1,1; |_ _ _ _| |_ _|_| _ _|
. 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1; |_ _ _ _ _| _| |
. 1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1; |_ _ _ _ _| | _|
. 1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1; |_ _ _ _ _ _| _ _|
. 1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1; |_ _ _ _ _ _| |
.1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1; |_ _ _ _ _ _ _|
...
Also consider the infinite double-staircases diagram defined in A335616. For n = 15 the diagram with first 15 levels looks like this:
.
Level "Double-staircases" diagram
. _
1 _|1|_
2 _|1 _ 1|_
3 _|1 |1| 1|_
4 _|1 _| |_ 1|_
5 _|1 |1 _ 1| 1|_
6 _|1 _| |1| |_ 1|_
7 _|1 |1 | | 1| 1|_
8 _|1 _| _| |_ |_ 1|_
9 _|1 |1 |1 _ 1| 1| 1|_
10 _|1 _| | |1| | |_ 1|_
11 _|1 |1 _| | | |_ 1| 1|_
12 _|1 _| |1 | | 1| |_ 1|_
13 _|1 |1 | _| |_ | 1| 1|_
14 _|1 _| _| |1 _ 1| |_ |_ 1|_
15 |1 |1 |1 | |1| | 1| 1| 1|
.
Starting from A196020 and after the algorithm described in A280850 and A296508 applied to the above diagram we have a new diagram as shown below: .
Level "Ziggurat" diagram
. _
6 |1|
7 _ | | _
8 _|1| _| |_ |1|_
9 _|1 | |1 1| | 1|_
10 _|1 | | | | 1|_
11 _|1 | _| |_ | 1|_
12 _|1 | |1 1| | 1|_
13 _|1 | | | | 1|_
14 _|1 | _| _ |_ | 1|_
15 |1 | |1 |1| 1| | 1|
.
The 15th row
of this seq: [1,1,1,1,1,1,1,1,0,0,0,1,1,1,2,1,1,1,0,0,0,1,1,1,1,1,1,1,1]
The 15th row
The 15th row
The 15th row
.
The number of horizontal steps (or 1's) in the successive columns of the above diagram gives the 15th row of this triangle.
For more information about the parts of the symmetric representation of sigma(n) see A237270. For more information about the subparts see A239387, A296508, A280851. More generally, it appears there is the same correspondence between the original diagram of the symmetric representation of sigma(n) and the "Ziggurat" diagram of n. (End)
MATHEMATICA
(* function segments are defined in A237270 *) a249351[n_] := Flatten[Map[segments, Range[n]]]
CROSSREFS
Cf. A000203, A003056, A067742, A071562, A165513, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A238443, A239660, A239932- A239934, A240542, A241008, A241010, A245092, A245685, A246955, A246956, A247687, A249223, A250068, A250070, A250071, A262626, A280850, A280851, A296508, A235616, A347186.
Numbers with no middle divisors (cf. A071090).
+10
44
3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 111, 113, 114
COMMENTS
Numbers k such that A071090(k) is 0. Conjecture: lim_{n->oo} a(n)/n = 4/3.
Regarding the above conjecture, numerical calculations suggest that this limit is smaller than 4/3. See A071540. - Amiram Eldar, Jul 27 2024 Also numbers n with the property that the number of parts in the symmetric representation of sigma(n) is even. - Michel Marcus and Omar E. Pol, Apr 25 2014 [For a proof see the link. - Hartmut F. W. Hoft, Sep 09 2015] Middle divisors are divisors d with sqrt(k/2) <= d < sqrt(2k). - Michael B. Porter, Oct 19 2018
EXAMPLE
The divisors of 21 are 1, 3, 7, and 21. Since none of these are between sqrt(21/2) = 3.24... and sqrt(2*21) = 6.48..., 21 is in the sequence.
The divisors of 20 are 1, 2, 4, 5, 10, and 20. Since 4 and 5 are both between sqrt(20/2) = 3.16... and sqrt(2*20) = 6.32..., 20 is not in the sequence. (End)
MATHEMATICA
f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[125], f[ # ] == 0 &]
(* Related to the symmetric representation of sigma *)
(* subsequence of even parts of number k for m <= k <= n *)
(* Function a237270[] is defined in A237270 *) (* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
a071561[m_, n_]:=Select[Range[m, n], EvenQ[Length[a237270[#]]]&]
a071561[1, 114] (* data *)
Select[Range@ 120, Function[n, Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] == {}]] (* Michael De Vlieger, Jan 03 2017 *)
PROG
(PARI) is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1 \\ Iain Fox, Dec 19 2017 (PARI) is(n, f=factor(n))=my(t=(n+1)\2); fordiv(f, d, if(d^2>=t, return(d^2>2*n))); 0 \\ Charles R Greathouse IV, Jan 22 2018 (PARI) list(lim)=my(v=List(), t); forfactored(n=3, lim\1, t=(n[1]+1)\2; fordiv(n[2], d, if(d^2>=t, if(d^2>2*n[1], listput(v, n[1])); break))); Vec(v) \\ Charles R Greathouse IV, Jan 22 2018
Maximum width of any region in the symmetric representation of sigma(n).
+10
27
1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2
COMMENTS
Since the width of the single region of the symmetric representation of sigma( 2^ceiling((p-1)*(log_2 3) - 1) * 3^(p-1) ), for prime number p, at the diagonal equals p, this sequence contains an increasing subsequence (see A250071). a(n) is also the number of layers of width 1 in the symmetric representation of sigma(n). For more information see A001227. - Omar E. Pol, Dec 13 2016
FORMULA
a(n) = max_{k=1..floor((sqrt(8*n+1) - 1)/2)} (Sum_{j=1..k}(-1)^(j+1)* A237048(n, j)), for n >= 1.
EXAMPLE
a(6) = 2 since the sequence of widths at each unit step in the symmetric representation of sigma(6) = 12 is 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1. For visual examples see A237270, A237593 and sequences referenced in these.
MATHEMATICA
(* function a2[ ] is defined in A249223 *) a250068[n_]:=Max[a2[n]]
a250068[{m_, n_}]:=Map[a250068, Range[m, n]]
a250068[{1, 100}](* data *)
PROG
(PARI) t237048(n, k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0);
kmax(n) = (sqrt(1+8*n)-1)/2;
t249223(n, k) = sum(j=1, k, (-1)^(j+1)*t237048(n, j));
a(n) = my(wm = t249223(n, 1)); for (k=2, kmax(n), wm = max(wm, t249223(n, k))); wm; \\ Michel Marcus, Sep 20 2015
CROSSREFS
Cf. A000203, A001227, A237048, A237270, A237271, A237591, A237593, A241008, A241010, A246955, A247687, A249223, A249351 (widths), A279387, A279388, A279391.
Smallest number k such that the symmetric representation of sigma(k) has at least one part of width n.
+10
27
1, 6, 60, 120, 360, 840, 3360, 2520, 5040, 10080, 15120, 32760, 27720, 50400, 98280, 83160, 110880, 138600, 221760, 277200, 332640, 360360, 554400, 960960, 831600, 942480, 720720, 2217600, 1965600, 1441440, 3160080, 2827440, 2162160, 2882880, 3603600, 5765760, 5654880, 4324320, 9979200
COMMENTS
The 26 entries starting with a(2) = 6 are products of powers of consecutive primes starting with 2, except for a(12) = 32760 and a(15) = 98280 (which are missing 11), and a(26) = 942480 (which is missing 13).
a(n) is the smallest number k such that the symmetric representation of sigma(k) has n layers. For more information see A279387. - Omar E. Pol, Dec 16 2016 All terms a(n) <= 1.75*10^7 have a symmetric representation of sigma that consists of a single part and they are abundant for n > 2. Numbers a(1) = 1, a(2) = 6, and a(4) = 120 are unimodal while numbers a(6) = 840, a(14) = 50400, a(18) = 138600, a(24) = 960960, a(26) = 942480, a(32) = 2827440, a(44) = 8648640 have a single extent of maximum width, but are not unimodal.
Conjecture: The symmetric representation of sigma for every term consists of a single part and it is unimodal only for a(1), a(2), and a(4).
As a consequence, this sequence would be a subsequence of A174973, and all a(n), n > 2, would be abundant. (End)
FORMULA
a(n) = min(k such that A250068(k) = n), n >= 1.
EXAMPLE
a(3) = 60 since the symmetric representation of sigma(60) = 168 consists of a single region of whose successive widths are 41 1's, 9 2's, 6 3's, 7 2's, 6 3's, 9 2's, and 41 1's.
a(6) = 840 has a single extent of 12 units of width 6 centered around point (583,583) on the diagonal, but is not unimodal. - Hartmut F. W. Hoft, Jun 10 2024
MATHEMATICA
(* function a2[ ] is defined in A249223 *) a250070[{j_, k_}, b_] := Module[{i, max, acc={{1, 1}}}, For[i=j, i<=k, i++, max={Max[a2[i]], i}; If[max[[1]]>b && !MemberQ[Transpose[acc][[1]], max[[1]]], AppendTo[acc, max]]]; acc]
(* returns (argument, result) data pairs since sequence is non-monotonic *)
Sort[a250070[{1, 1000000}, 1]] (* computed in steps *)
CROSSREFS
Cf. A000203, A237270, A237271, A237593, A241008, A241010, A246955, A247687, A249223, A250071, A253258, A279387.
Numbers j for which the symmetric representation of sigma(j) has two parts, each of width one.
+10
21
3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 92, 94, 97, 101, 103, 106, 107, 109, 113, 116, 118, 122, 124, 127, 131, 134, 136, 137, 139, 142, 146, 148, 149, 151, 152, 157, 158, 163, 164, 166, 167, 172, 173, 178, 179, 181, 184, 188, 191, 193, 194, 197, 199
COMMENTS
The sequence is the intersection of A239929 (sigma(j) has two parts) and of A241008 (sigma(j) has an even number of parts of width one). The numbers in the sequence are precisely those defined by the formula for the triangle, see the link. The symmetric representation of sigma(j) has two parts, each part having width one, precisely when j = 2^(k - 1) * p where 2^k <= row(j) < p, p is prime and row(j) = floor((sqrt(8*j + 1) - 1)/2). Therefore, the sequence can be written naturally as a triangle as shown in the Example section.
The symmetric representation of sigma(j) = 2*j - 2 consists of two regions of width 1 that meet on the diagonal precisely when j = 2^(2^m - 1)*(2^(2^m) + 1) where 2^(2^m) + 1 is a Fermat prime (see A019434). This subsequence of numbers j is 3, 10, 136, 32896, 2147516416, ...[?]... ( A191363). The k-th column of the triangle starts in the row whose initial entry is the first prime larger than 2^(k+1) (that sequence of primes is A014210, except for 2). Observation: at least the first 82 terms coincide with the numbers j with no middle divisors whose largest divisor <= sqrt(j) is a power of 2, or in other words, coincide with the intersection of A071561 and A365406. - Omar E. Pol, Oct 11 2023
FORMULA
Formula for the triangle of numbers associated with the sequence:
P(n, k) = 2^k * prime(n) where n >= 2, 0 <= k <= floor(log_2(prime(n)) - 1).
EXAMPLE
We show portions of the first eight columns, 0 <= k <= 7, of the triangle.
0 1 2 3 4 5 6 7
3
5 10
7 14
11 22 44
13 26 52
17 34 68 136
19 38 76 152
23 46 92 184
29 58 116 232
31 62 124 248
37 74 148 296 592
41 82 164 328 656
43 86 172 344 688
47 94 188 376 752
53 106 212 424 848
59 118 236 472 944
61 122 244 488 976
67 134 268 536 1072 2144
71 142 284 568 1136 2272
. . . . . .
. . . . . .
127 254 508 1016 2032 4064
131 262 524 1048 2096 4192 8384
137 274 548 1096 2192 4384 8768
. . . . . . .
. . . . . . .
251 502 1004 2008 4016 8032 16064
257 514 1028 2056 4112 8224 16448 32896
263 526 1052 2104 4208 8416 16832 33664
Since 2^(2^4) + 1 = 65537 is the 6543rd prime, column k = 15 starts with 2^15*(2^(2^16) + 1) = 2147516416 in row 6542 with 65537 in column k = 0.
For an image of the symmetric representations of sigma(m) for all values m <= 137 in the triangle see the link.
The first column is the sequence of odd primes, see A065091. The second column is the sequence of twice the primes starting with 10, see A001747. The third column is the sequence of four times the primes starting with 44, see A001749. For related references also see A033676 (largest divisor of n less than or equal to sqrt(n)).
MATHEMATICA
(* functions path[] and a237270[ ] are defined in A237270 *) atmostOneDiagonalsQ[n_]:=SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], - 1] - path[n - 1], 1]]]
(* data *)
Select[Range[200], Length[a237270[#]]==2 && atmostOneDiagonalsQ[#]&]
(* function for computing triangle in the Example section through row 55 *)
TableForm[Table[2^k Prime[n], {n, 2, 56}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth->2]
CROSSREFS
Cf. A000203, A033676, A071561, A163280, A237270, A237271, A237593, A241008, A241010, A247687, A250068, A250070, A250071, A365406.
Numbers m with the property that the symmetric representation of sigma(m) has three parts of width one.
+10
19
9, 25, 49, 50, 98, 121, 169, 242, 289, 338, 361, 484, 529, 578, 676, 722, 841, 961, 1058, 1156, 1369, 1444, 1681, 1682, 1849, 1922, 2116, 2209, 2312, 2738, 2809, 2888, 3362, 3364, 3481, 3698, 3721, 3844, 4232, 4418, 4489, 5041, 5329, 5476, 5618, 6241, 6724, 6728, 6889, 6962, 7396, 7442, 7688, 7921, 8836, 8978, 9409
COMMENTS
The symmetric representation of sigma(m) has 3 regions of width 1 where the two extremal regions each have 2^k - 1 legs and the central region starts with the p-th leg of the associated Dyck path for sigma(m) precisely when m = 2^(k - 1) * p^2 where 2^k < p <= row(m), k >= 1, p >= 3 is prime and row(m) = floor((sqrt(8*m + 1) - 1)/2). Furthermore, the areas of the two outer regions are (2^k - 1)*(p^2 + 1)/2 each so that the area of the central region is (2^k - 1)*p; for a proof see the link.
Since the sequence is defined by a two-parameter expression it can be written naturally as a triangle as shown in the Example section.
A263951 is a subsequence of this sequence containing the squares of all those primes p for which the areas of the 3 regions in the symmetric representation of p^2 (p once and (p^2 + 1)/2 twice) are primes; i.e., p^2 and p^2 + 1 are semiprimes (see A070552). - Hartmut F. W. Hoft, Aug 06 2020
FORMULA
As an irregular triangle, T(n, k) = 2^k * prime(n)^2 where n >= 2 and 0 <= k <= floor(log_2(prime(n)) - 1).
EXAMPLE
We show portions of the first eight columns, powers of two 0 <= k <= 7, and 55 rows of the triangle through prime(56) = 263.
p/k 0 1 2 3 4 5 6 7
3 9
5 25 50
7 49 98
11 121 242 484
13 169 338 676
17 289 578 1156 2312
19 361 722 1444 2888
23 529 1058 2116 4232
29 841 1682 3364 6728
31 961 1922 3844 7688
37 1369 2738 5476 10952 21904
41 1681 3362 6724 13448 26896
43 1849 3698 7396 14792 29584
47 2209 4418 8836 17672 35344
53 2809 5618 11236 22472 44944
59 3481 6962 13924 27848 55696
61 3721 7442 14884 29768 59536
67 4489 8978 17956 35912 71824 143648
71 5041 10082 20164 40328 80656 161312
. . . . . . .
. . . . . . .
131 17161 34322 68644 137288 274567 549152 1098304
137 18769 37538 75076 150152 300304 600608 1201216
. . . . . . . .
. . . . . . . .
257 66049 132098 264196 528392 1056784 2113568 4227136 8454272
263 69169 138338 276676 553352 1106704 2213408 4426816 8853632
Number 4 is not in this sequence since the symmetric representation of sigma(4) consists of a single region. Column k=0 contains the squares of primes ( A001248(n), n>=2), column k=1 contains double the squares of primes ( A079704(n), n>=2) and column k=2 contains four times the squares of primes ( A069262(n), n>=5).
MATHEMATICA
(* path[n] and a237270[n] are defined in A237270 *) atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
(* data *)
Select[Range[10000], atmostOneDiagonalsQ[#] && Length[a237270[#]]==3 &]
(* expression for the triangle in the Example section *)
TableForm[Table[2^k Prime[n]^2, {n, 2, 57}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth -> 2, TableHeadings -> {Map[Prime, Range[2, 57]], Range[0, Floor[Log[2, Prime[57] - 1]]]}]
Numbers with no pair (d,e) of divisors such that d < e < 2*d.
+10
17
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 101, 103, 106
COMMENTS
sequences of powers of primes are subsequences;
In other words: numbers n with the property that all parts in the symmetric representation of sigma(n) have width 1. - Omar E. Pol, Dec 08 2016 Sequence A357581 shows the numbers organized in columns of a square array by the number of parts in their symmetric representation of sigma. - Hartmut F. W. Hoft, Oct 04 2022
MAPLE
filter:= proc(n)
local d, q;
d:= numtheory:-divisors(n);
min(seq(d[i+1]/d[i], i=1..nops(d)-1)) >= 2
end proc:
MATHEMATICA
(* it suffices to test adjacent divisors *)
a174905[n_] := Module[{d = Divisors[n]}, ! Apply[Or, Map[2 #[[1]] > #[[2]] &, Transpose[{Drop[d, -1], Drop[d, 1]}]]]]
Select[Range[106], !MatchQ[Divisors[#], {___, d_, e_, ___} /; e < 2d]& ] (* Jean-François Alcover, Jan 31 2018 *)
PROG
(Haskell)
a174905 n = a174905_list !! (n-1)
a174905_list = filter ((== 0) . a174903) [1..]
CROSSREFS
Cf. A000040, A000961, A001248, A005279, A030078, A030514, A129511, A174903, A237271, A237593, A241008, A241010.
Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd, and that all parts have width 1.
+10
16
1, 2, 4, 8, 9, 16, 25, 32, 49, 50, 64, 81, 98, 121, 128, 169, 242, 256, 289, 338, 361, 484, 512, 529, 578, 625, 676, 722, 729, 841, 961, 1024, 1058, 1156, 1250, 1369, 1444, 1681, 1682, 1849, 1922, 2048, 2116, 2209, 2312, 2401, 2738, 2809, 2888, 3025, 3249, 3362, 3364, 3481, 3698, 3721, 3844
COMMENTS
The first eight entries in A071562 but not in this sequence are 6, 12, 15, 18, 20, 24, 28 & 30. The first eight entries in A238443 but not in this sequence are 6, 12, 18, 20, 24, 28, 30 & 36. Let n = 2^m * product(p_i^e_i, i=1,...,k) = 2^m * q with m >= 0, k >= 0, 2 < p_1, ...< p_k primes and e_i >= 1, for all 1 <= i <= k. For each number n in this sequence all e_i are even, and for any two odd divisors f < g of n, 2^(m+1) * f < g. The sum of the areas of the regions r(n, z) equals sigma(n). For a proof of the characterization and the formula see the theorem in the link below.
Numbers 3025 = 5^2 * 11^2 and 510050 = 2^1 * 5^2 * 101^2 are the smallest odd and even numbers, respectively, in the sequence with two distinct odd prime factors.
Among the 706 numbers in the sequence less than 1000000 (see link to the table) there are 143 that have two different odd prime factors, but none with three. All numbers with three different odd prime factors are larger than 500000000. Number 4450891225 = 5^2 * 11^2 * 1213^2 is in the sequence, but may not be the smallest one with three different odd prime factors. Note that 1213 is the first prime that extends the list of divisors of 3025 while preserving the property for numbers in this sequence.
The subsequence of numbers n = 2^(k-1) * p^2 satisfying the constraints above is A247687. n = 3^(2*h) = 9^h = A001019(h), h>=0, is the smallest number for which the symmetric representation of sigma(n) has 2*h+1 regions of width one, for example for h = 1, 2, 3 and 5, but not for h = 4 in which case 3025 = 5^2 * 11^2 < 3^8 = 6561 is the smallest (see A318843). [Comment corrected by Hartmut F. W. Hoft, Sep 04 2018] Computations using this characterization are more efficient than those of Dyck paths for the symmetric representations of sigma(n), e.g., the Mathematica code below.
FORMULA
Formula for the z-th region in the symmetric representation of n = 2^m * q in this sequence, 1 <= z <= sigma_0(q) and q odd: r(n, z) = 1/2 * (2^(m+1) - 1) * (d_z + d_(2*x+2-z)) where 1 = d_1 < ... < d_(2*x+1) = q are the odd divisors of n.
EXAMPLE
This irregular triangle presents in each column those elements of the sequence that have the same factor of a power of 2.
row/col 2^0 2^1 2^2 2^3 2^4 2^5 ...
2^k: 1 2 4 8 16 32 ...
3^2: 9
5^2: 25 50
7^2: 49 98
3^4: 81
11^2: 121 242 484
13^2: 169 338 676
17^2: 289 578 1156 2312
19^2: 361 722 1444 2888
23^2: 529 1058 2116 4232
5^4: 625 1250
3^6: 729
29^2: 841 1682 3364 6728
31^2: 961 1922 3844 7688
37^2: 1369 2738 5476 10952 21904
41^2: 1681 3362 6724 13448 26896
43^2: 1849 3698 7396 14792 29584
47^2: 2209 4418 8836 17672 35344
7^4: 2401 4802
53^2: 2809 5618 11236 22472 44944
5^2*11^2: 3025
3^2*19^2: 3249
59^2: 3481 6962 13924 27848 55696
61^2: 3721 7442 14884 29768 59536
67^2: 4489 8978 17956 35912 71824 143648
3^2*23^2: 4761
71^2: 5041
...
5^2*101^2:225025 510050
...
Number 3025 = 5^2 * 11^2 is in the sequence since its divisors are 1, 5, 11, 25, 55, 121, 275, 605 and 3025. Number 6050 = 2^1 * 5^2 * 11^2 is not in the sequence since 2^2 * 5 > 11 while 5 < 11.
Number 510050 = 2^1 * 5^2 * 101^2 is in the sequence since its 9 odd divisors 1, 5, 25, 101, 505, 2525, 10201, 51005 and 225025 are separated by factors larger than 2^2. The areas of its 9 regions are 382539, 76515, 15339, 3939, 1515, 3939, 15339, 76515 and 382539. However, 2^2 * 5^2 * 101^2 is not in the sequence.
The rows, except the first, are indexed by products of even powers of the odd primes satisfying the property, sorted in increasing order.
The first column is a subsequence of A244579. A row labeled p^(2*h), h>=1 and p>=3 with p = A000040(n), has A098388(n) entries. Starting with the second column, dividing the entries of a column by 2 creates a proper subsequence of the prior column.
See A259417 for references to other sequences of even powers of odd primes that are subsequences of column 1. The first entry greater than 16 in column labeled 2^4 is 21904 since 37 is the first prime larger than 2^5. The rightmost entry in the row labeled 19^2 is 2888 in the column labeled 2^3 since 2^4 < 19 < 2^5.
MATHEMATICA
(* path[n] and a237270[n] are defined in A237270 *) atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
Select[Range[1000], atmostOneDiagonalsQ[#] && OddQ[Length[a237270[#]]]&] (* data *)
(* more efficient code based on numeric characterization *)
divisorPairsQ[m_, q_] := Module[{d = Divisors[q]}, Select[2^(m + 1)*Most[d] - Rest[d], # >= 0 &] == {}]
a241010AltQ[n_] := Module[{m, q, p, e}, m=IntegerExponent[n, 2]; q=n/2^m; {p, e} = Transpose[FactorInteger[q]]; q==1||(Select[e, EvenQ]==e && divisorPairsQ[m, q])]
a241010Alt[m_, n_] := Select[Range[m, n], a241010AltQ]
a241010Alt[1, 4000] (* data *)
CROSSREFS
Cf. A000203, A174905, A236104, A237270 (symmetric representation of sigma(n)), A237271, A237593, A238443, A241008, A071562, A246955, A247687, A250068, A250070, A250071.
EXTENSIONS
More terms and further edited by Hartmut F. W. Hoft, Jun 26 2015 and Jul 02 2015 and corrected Oct 11 2015
Smallest number k such that the symmetric representation of sigma(k) has maximum width n for those k whose representation has nondecreasing width up to the diagonal.
+10
16
1, 6, 72, 120, 5184, 1440, 373248, 6720, 28800, 103680, 1934917632, 80640, 278628139008, 7464960, 2073600, 483840, 1444408272617472, 1612800, 103997395628457984, 5806080
COMMENTS
The symmetric representation of sigma(k) has nondecreasing width to the diagonal precisely when all odd divisors counted in the k-th row of A237048 occur at odd indices. If we write k = 2^m * q with m >= 0 and q odd, this property is equivalent to q < 2^(m+1). The values for a(11), a(13), a(17) and a(19) were computed directly using the formula k = 2^m * 3^(p-1) where p is one of the four primes and m the smallest exponent so that 3^(p-1) < 2^(m+1). Each of these numbers has a symmetric representation of nondecreasing width ending in a prime number width, and they are the first such numbers since the number of divisors of an odd number is a prime precisely when the number is a power of a prime.
The other numbers listed whose symmetric representations of sigma(k) have nondecreasing width are smaller than 7500000. The only additional numbers k <= 100000000 are a(24) = 7096320, a(27) = 90316800 and a(32) = 85155840.
FORMULA
a(n) = min(2^m * q, m >= 0 & q odd & sigma_0(q) = n & q < 2^(m+1)) where sigma_0 is the number of divisors.
a(p) = 2^ceiling((p-1)*(log_2(3)) - 1) * 3^(p-1) for primes p.
EXAMPLE
a(6) = 1440 = 2^5 * 3^2 * 5 has 6 odd divisors. It is the smallest number of the form 2^m * q with m > 0, q odd and such that q < 2^(m+1).
MATHEMATICA
(* function a2[ ] is defined in A249223 *) smallQ[n_] := Module[{x=2^IntegerExponent[n, 2]}, n/x<2x]
ndWidth[{m_, n_}] := Select[Range[m, n], smallQ]
a250071[x_List] := Module[{i, max, acc={{1, 1}}}, For[i=1, i<=Length[x], i++, max={Max[a2[x[[i]]]], x[[i]]}; If[!MemberQ[Transpose[acc][[1]], max[[1]]], AppendTo[acc, max]]]; acc]
(* returns (argument, result) data pairs since sequence is non-monotonic *)
Sort[a250071[ndWidth[{1, 100000000}]]] (* computed in steps *)
CROSSREFS
Cf. A000203, A237048, A237270, A237271, A237593, A241008, A241010, A246955, A247687, A249223, A250068, A250070.
Triangle read by rows in which row n lists the widths of the symmetric representation of A024916(n): the sum of all divisors of all positive integers <= n.
+10
9
1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 3, 2, 1, 1, 2, 3, 3, 3, 3, 3, 2, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 4, 5, 4, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 6, 5, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 6, 7, 6, 5, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 6, 7, 7, 7, 6, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6, 5, 4, 3, 2, 1
COMMENTS
Here T(n,k) is defined to be the "k-th width" of the symmetric representation of A024916(n), with n>=1 and 1<=k<=2n-1. If both A249351 and this sequence are written as isosceles triangles then the partial sums of the columns of A249351 give the columns of this isosceles triangle (see the second triangle in Example section). For the definition of the k-th width of the symmetric representation of sigma(n) see A249351. Note that for the geometric representation of the n-th row of the triangle we need the x-axis, the y-axis, and only a Dyck path which is given by the elements of the n-th row of the triangle A237593. Row n has length 2*n-1.
EXAMPLE
Triangle begins:
1;
1,2,1;
1,2,2,2,1;
1,2,3,3,3,2,1;
1,2,3,3,3,3,3,2,1;
1,2,3,4,4,5,4,4,3,2,1;
1,2,3,4,4,4,5,4,4,4,3,2,1;
1,2,3,4,5,5,5,6,5,5,5,4,3,2,1;
1,2,3,4,5,5,5,6,7,6,5,5,5,4,3,2,1;
1,2,3,4,5,6,6,6,7,7,7,6,6,6,5,4,3,2,1;
1,2,3,4,5,6,6,6,6,7,7,7,6,6,6,6,5,4,3,2,1;
1,2,3,4,5,6,7,7,7,8,9,9,9,8,7,7,7,6,5,4,3,2,1;
...
--------------------------------------------------------------------------
. Written as an isosceles triangle
. the sequence begins: Diagram for n = 1..12
--------------------------------------------------------------------------
. _ _ _ _ _ _ _ _ _ _ _ _
. 1; |_| | | | | | | | | | | |
. 1,2,1; |_ _|_| | | | | | | | | |
. 1,2,2,2,1; |_ _| _|_| | | | | | | |
. 1,2,3,3,3,2,1; |_ _ _| _|_| | | | | |
. 1,2,3,3,3,3,3,2,1; |_ _ _| _| _ _|_| | | |
. 1,2,3,4,4,5,4,4,3,2,1; |_ _ _ _| _| | _ _|_| |
. 1,2,3,4,4,4,5,4,4,4,3,2,1; |_ _ _ _| |_ _|_| _ _|
. 1,2,3,4,5,5,5,6,5,5,5,4,3,2,1; |_ _ _ _ _| _| |
. 1,2,3,4,5,5,5,6,7,6,5,5,5,4,3,2,1; |_ _ _ _ _| | _|
. 1,2,3,4,5,6,6,6,7,7,7,6,6,6,5,4,3,2,1; |_ _ _ _ _ _| _ _|
. 1,2,3,4,5,6,6,6,6,7,7,7,6,6,6,6,5,4,3,2,1; |_ _ _ _ _ _| |
.1,2,3,4,5,6,7,7,7,8,9,9,9,8,7,7,7,6,5,4,3,2,1; |_ _ _ _ _ _ _|
...
For n = 3 the symmetric representation of A024916(3) = 8 in the 4th quadrant looks like this: .
. Polygon Cells
. _ _ _ _ _ _
. | | |_|_|_|
. | _| |_|_|_|
. |_ _| |_|_|
.
There are eight cells. The representation of the widths looks like this:
.
. \ \ \
. \ \ \
. \ \ 1
. 2 2
. 1 2
.
So the third row of the triangle is [1, 2, 2, 2, 1].
CROSSREFS
Cf. A024916, A236104, A237048, A237593, A240542, A241008, A241010, A244250, A245685, A246955, A247687, A249351, A250068, A250070, A250071, A253258, A262626.
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