Displaying 11-20 of 23 results found.
Composite numbers n with the property that the symmetric representation of sigma(n) has two parts.
+10
5
10, 14, 22, 26, 34, 38, 44, 46, 52, 58, 62, 68, 74, 76, 78, 82, 86, 92, 94, 102, 106, 114, 116, 118, 122, 124, 134, 136, 138, 142, 146, 148, 152, 158, 164, 166, 172, 174, 178, 184, 186, 188, 194, 202, 206, 212, 214, 218, 222, 226, 232, 236, 244, 246, 248, 254, 258, 262, 268, 274, 278, 282, 284, 292, 296, 298, 302, 314, 316, 318, 326, 328, 332, 334, 344, 346, 348, 354, 356, 358
COMMENTS
By definition the two parts of the symmetric representation of sigma(n) are sigma(n)/2 and sigma(n)/2.
EXAMPLE
Illustration of the symmetric representation of sigma(n) in the second quadrant for the first four elements of this sequence: [10, 14, 22, 26].
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n: 26 22 14 10
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Sigma(10) = 9 + 9 = 18.
Sigma(14) = 12 + 12 = 24.
Sigma(22) = 18 + 18 = 36.
Sigma(26) = 21 + 21 = 42.
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CROSSREFS
Cf. A237271 (number of parts), A237270, A237593, A238443, A238524, A239660, A239929, A239932, A239934, A245092, A262626, A280107 (4 parts).
Numbers k with the property that the symmetric representation of sigma(k) has six parts.
+10
5
147, 171, 189, 207, 243, 261, 275, 279, 297, 333, 351, 363, 369, 387, 423, 429, 465, 477, 507, 531, 549, 555, 595, 603, 605, 615, 639, 645, 657, 663, 705, 711, 715, 741, 747, 795, 801, 833, 845, 867, 873, 885, 909, 915, 927, 931, 935, 963, 969, 981, 1005, 1017, 1045, 1065, 1071, 1083, 1095, 1105, 1127
COMMENTS
Those numbers in this sequence with only parts of width 1 in their symmetric representation of sigma form column 6 in the table of A357581. - Hartmut F. W. Hoft, Oct 04 2022
EXAMPLE
147 is in the sequence because the 147th row of A237593 is [74, 25, 13, 8, 5, 4, 4, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 4, 4, 5, 8, 13, 25, 74], and the 146th row of the same triangle is [74, 25, 12, 8, 6, 4, 3, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 4, 6, 8, 12, 25, 74], therefore between both symmetric Dyck paths there are six parts: [74, 26, 14, 14, 26, 74]. Note that the sum of these parts is 74 + 26 + 14 + 14 + 26 + 74 = 228, equaling the sum of the divisors of 147: 1 + 3 + 7 + 21 + 49 + 147 = 228.
(The diagram of the symmetric representation of sigma(147) = 228 is too large to include.)
MATHEMATICA
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
a320511[n_] := Select[Range[n], partsSRS[#]==6&]
CROSSREFS
Cf. A000203, A018303, A196020, A235791, A236104, A237048, A237591, A237593, A239663, A239665, A245092, A262626, A296508.
Numbers n for which the symmetric representation of sigma(n) has at least 3 parts, all having the same area.
+10
3
EXAMPLE
The parts of the symmetric representations of sigma(15) and sigma(5950) are {8, 8, 8} and {4464, 4464, 4464}, respectively, so a(1) = 15 and a(2) = 5950.
Illustration of the symmetric representation of sigma(15) = 8 + 8 + 8 = 24 in the first quadrant:
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The three parts have the same area.
(End)
MATHEMATICA
(* T[], row[], cD[] & tD[] are defined in A239663 *) a251820[n_] := Module[{pT = T[n, 1], cT, cL, cW = 0, cR = 0, sects = {}, j = 1, r = row[n], test = True}, While[test && j <= r, cT = T[n, j+1]; cL = pT - cT; cW += (-1)^(j+1) * tD[n, j]; If[cW == 0 && cR != 0, AppendTo[sects, cR]; cR = 0; If[Min[sects] != Max[sects], test = False], cR += cL * cW]; pT = cT; j++]; If[cW != 0, AppendTo[sects, 2 * cR - cW]]; Min[sects] == Max[sects] && Length[sects] > 1]
Select[Range[50000], a251820] (* data *)
a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.
+10
3
2, 10, 50, 230, 1150, 5050, 22310, 106030, 510050, 2065450, 10236350
COMMENTS
It appears that a(n) = 2 * q where q is odd and that the symmetric representation of sigma(a(n)/2) has the same number of parts as that for a(n). Number a(12) > 15000000. - Hartmut F. W. Hoft, Sep 22 2021
EXAMPLE
a(1) = 2 because the second row of A237593 is [2, 2], and the first row of the same triangle is [1, 1], therefore between both symmetric Dyck paths there is only one part: [3], equaling the sum of the divisors of 2: 1 + 2 = 3. See below: .
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a(2) = 10 because the 10th row of A237593 is [6, 2, 1, 1, 1, 1, 2, 6], and the 9th row of the same triangle is [5, 2, 2, 2, 2, 5], therefore between both symmetric Dyck paths there are two parts: [9, 9]. Also there are no even numbers k < 10 whose symmetric representation of sigma(k) has two parts. Note that the sum of these parts is 9 + 9 = 18, equaling the sum of the divisors of 10: 1 + 2 + 5 + 10 = 18. See below: .
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a(3) = 50 because the 50th row of A237593 is [26, 9, 4, 3, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 3, 4, 9, 26], and the 49th row of the same triangle is [25, 9, 5, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 5, 9, 25], therefore between both symmetric Dyck paths there are three parts: [39, 15, 39]. Also there are no even numbers k < 50 whose symmetric representation of sigma(k) has three parts. Note that the sum of these parts is 39 + 15 + 39 = 93, equaling the sum of the divisors of 50: 1 + 2 + 5 + 10 + 25 + 50 = 93. (The diagram of the symmetric representation of sigma(50) = 93 is too large to include.)
MATHEMATICA
a320521[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=2, i<=n, i+=2, v=Count[a341969[i], 0]+1; If[list[[v]]==0, list[[v]]=i]]; list]
CROSSREFS
Cf. A000203, A018262, A005843, A196020, A235791, A236104, A237048, A237591, A237593, A239663, A239665, A240062, A245092, A262626, A296508.
Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.
+10
3
2, 4, 10, 6, 14, 50, 8, 22, 70, 230, 12, 26, 98, 250, 1150, 16, 34, 110, 290, 1250, 5050, 18, 38, 130, 310, 1450, 5150, 22310, 20, 44, 154, 370, 1550, 5290, 23230, 106030, 24, 46, 170, 406, 1850, 5350, 23690, 106490, 510050, 28, 52, 182, 410, 2030, 5450, 24610, 107410, 513130, 2065450
COMMENTS
This is a permutation of the positive even numbers ( A299174). The union of all odd-indexed columns gives A319796, the even numbers in A071562. The union of all even-indexed columns gives A319802, the even numbers in A071561.
EXAMPLE
The 10x10 section of table T(n,k):
------------------------------------------------------------------
n\k | 1 2 3 4 5 6 7 8 9 10 ...
------------------------------------------------------------------
1 | 2 10 50 230 1150 5050 22310 106030 510050 2065450
2 | 4 14 70 250 1250 5150 23230 106490 513130 2115950
3 | 6 22 98 290 1450 5290 23690 107410 520150 2126050
4 | 8 26 110 310 1550 5350 24610 110170 530150 2157850
5 | 12 34 130 370 1850 5450 25070 112010 530450 2164070
6 | 16 38 154 406 2030 5650 25250 112930 532450 2168150
7 | 18 44 170 410 2050 5750 25750 114770 534290 2176550
8 | 20 46 182 430 2150 6250 25990 115690 537050 2186650
9 | 24 52 190 434 2170 6350 26450 116150 540350 2216950
10| 28 58 238 470 2350 6550 26750 117070 544870 2219650
... (End)
MATHEMATICA
(* function a341969 is defined in A341969 *) sArray[b_, pMax_] := Module[{list=Table[{}, pMax], i, p}, For[i=2, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax&&Length[list[[p]]]<pMax, AppendTo[list[[p]], i]]]; list]
rank[n_] := n-row[n-1](row[n-1]+1)/2
parts[n_] := row[n-1]-rank[n]+2
a320537[sMatrix_, aD_] := Map[sMatrix[[rank[#], parts[#]]]&, Range[aD (aD+1)/2]]/; MatrixQ[sMatrix]&&aD<=Length[sMatrix]
m2500000=sArray[2500000, 10] (* entire 10x10 matrix needs to be computed *)
a320537[m2500000, 10] (* Sequence Data a(1..55) *)
CROSSREFS
Cf. A000203, A071561, A071562, A236104, A237270, A237271, A237593, A238443, A239663, A239665, A239929, A240062, A245092, A262626, A299174, A319796, A319802, A341969, A341970, A341971, A346969, A348171.
Numbers k with the property that the parts of the symmetric representation of sigma(k) are two octagons.
+10
3
10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526
COMMENTS
Note that odd primes ( A065091) are also the numbers j with the property that the parts of the symmetric representation of sigma(j) are two rectangles or more generally two quadrilaterals. Conjecture 1: The octagons are S-shaped and they have width 1.
Conjecture 2: This sequence is also the primes doubled (or even semiprimes) >= 10 (Cf. A100484). - Omar E. Pol, Aug 15 2023 For the symmetric representation of sigma(n) to consist of 2 octagons the first 3 entries in row n of the triangle of A249223 must be nonzero, hence must be 1's, indicating width 1, with the remaining entries zero. Therefore, row n of A237048 is 100100..., implying n = 2*p with p>3 prime. Both conjectures are true. - Hartmut F. W. Hoft, Aug 22 2023 Also the row numbers of the triangle A364639 where the rows are [1, 0, -1, 1] or where the rows start with [1, 0, -1, 1] and the remaining terms are zeros. Each supersequence A063221 >= 10 and A091999 >= 10 gives the numbers k with the property that the first part of the symmetric representation of sigma(k) is an octagon. In that case each supersequence gives the row numbers of the triangle A364639 where the rows start with [1, 0, -1]. (End)
EXAMPLE
The symmetric representation of sigma(14) in the first quadrant is as follows:
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The diagram has only two parts (or polygons) and both are octagons so 14 is in the sequence.
CROSSREFS
Cf. A000203, A001747, A065091, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A241008, A245092, A246955, A249223, A249351, A262626, A364414, A364639, A365081.
a(n), n>1, is the smallest number k whose symmetric representation of sigma(k) has two parts and has a larger number of legs in its two parts than a(n-1); a(1)=3.
+10
2
3, 10, 44, 78, 136, 348, 592, 666, 820, 1272, 1652, 1830, 2144, 2628, 3320, 3738, 4656, 5886, 6328, 7620, 8384, 9042, 10728, 13040, 14532, 15752, 16290, 18528, 21100, 21944, 24084, 25424, 28920, 32382, 32896, 35508, 39340, 42192, 46050, 48828
COMMENTS
A number k with two parts in its symmetric representation of sigma(k) [ssrs(k) = 2] has the form k = q*p with q in A174973, p prime and 2*q < p. This implies that 2*q <= row(k) < p and the first 0 in the k-th row of A249223 (having row(k) = floor((sqrt(8*k+1)-1)/2) entries) occurs at position 2*q so that 2*q-1 is the number of legs in each of the two parts. Therefore, the numbers 2*q-1 with q in A174973 are the only possible leg counts when ssrs(k) = 2, and for given q in A174973 and smallest prime p(q) > 2*q the number k = q*p(q) is the smallest with a leg count of 2*q-1. Consequently, each number q*p in the column of the irregular triangle A239929 labeled by q in A174793 with p prime satisfies ssrs(q*p) = 2*q-1. a(1) = 3 is the only odd number since 1 is the only odd number in A174973. Every number n = 2^m * p, m >= 0, 2^(m+1) < p and p prime, in this sequence is the sum of 2^(m+1) consecutive positive integers which includes every number in A246956.
EXAMPLE
a(3)=44 is the smallest number whose symmetric representation has 2 parts and 7 legs in each part.
a(4)=78 is the smallest number whose symmetric representation has 2 parts and 11 legs in each part.
No number k whose symmetric representation of sigma(k) has 2 parts can have 21 legs in its parts since there is no q in A174973 such that 2*q - 1 = 21.
MATHEMATICA
a174973Q[n_] := Module[{d=Divisors[n]}, Select[Rest[d] - 2*Most[d], #>0&]=={}]
a279105[n_] := Map[# * NextPrime[2*#]&, Select[Range[n], a174973Q]]
a279105[150] (* sequence data *)
One half of composite numbers k with the property that the symmetric representation of sigma(k) has two parts.
+10
2
5, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 86, 87, 89, 92, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 116, 118, 122, 123, 124, 127, 129, 131, 134, 137, 139, 141, 142, 146, 148, 149, 151, 157, 158, 159, 163, 164
COMMENTS
Also, even numbers of A239929 divided by two.
EXAMPLE
5 is in the sequence because 10 is a composite number, and the symmetric representation of sigma(10) = 18 has two parts (as shown below), and 10/2 = 5.
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CROSSREFS
Cf. A101550, A237271 (number of parts), A237270, A237593, A238443, A238524, A239929 (two parts), A239660, A239929, A239932, A239934, A240062 (k parts), A244894, A245092, A262626, A280107 (four parts).
a(n) = k if the binary representation of k has a 1 (0) exactly where a 1 in the n-th row of A237048 occurs at an odd (even) position, reading from left to right.
+10
2
1, 1, 2, 1, 2, 3, 2, 1, 5, 2, 2, 3, 2, 2, 11, 1, 2, 6, 2, 3, 10, 2, 2, 3, 5, 2, 10, 3, 2, 13, 2, 1, 10, 2, 11, 6, 2, 2, 10, 3, 2, 13, 2, 2, 45, 2, 2, 3, 5, 5, 10, 2, 2, 13, 10, 3, 10, 2, 2, 14, 2, 2, 43, 1, 10, 13, 2, 2, 10, 11, 2, 7, 2, 2, 44, 2, 11, 12, 2, 3, 21, 2, 2, 14, 10
COMMENTS
The phrase "symmetric representation of sigma(n)" is abbreviated below as SRS(n).
Every number in this sequence is a nondiving number and therefore in A061854. Number 22 with binary pattern 10110 is the smallest nondiving number in A061854, but not in this sequence since a number n with 5 odd divisors must have the form n = 2^m * p^4 for some prime p and some m>=0, and the pattern 10110 of odd/even positions of 1's in a row of A237048 requires 1's at positions 1 < 2^(m+1) < p < p^2 < 2^(m+1) * p <= row(n), a contradiction. a(2^n) = 1 for all n>=0. The single part of SRS(2^n) has width 1, see A238443. a(2^m * p) = 3 for odd primes p < 2^(m+1) with m >= 1. SRS(2^m * p) consists of a single part whose 2 subparts have sizes 2*T(n, 1) - 1 = 2^m * p - 1 and 2*T(n, p) - 1 = 2^m - p where T(n, k) = ceiling((n+1)/k -(k+1)/2), see A235791. The numbers 2^m * p are a subsequence of A174973 = A238443. a(p^k) = A000975(k+1) for all odd primes p and k >= 0. Number a(p^k) in binary has k+1 digits with 1's and 0's alternating. SRS(p^k) has k+1 parts all of width 1 and of the symmetric sizes T(p^k, p^i) - T(p^k, 2*p^i) = (p^(k-i) + p^i)/2, for 0 <= i <= k. The numbers p^k are a subsequence of A174905, the odd primes p form the 1st column in the irregular triangle of A239929 and the numbers p^2 form the 1st column in the irregular triangle of A247687.
EXAMPLE
Sequence values for the first 4 powers of 3: {a(1), a(3), a(9), a(27)} = {1, 2, 5, 10} = {1, 10, 101, 1010}.
Table for a(1..16), a(27) and a(28) together with their lists of the base-2 representation, of the odd/even positions of 1's in the n-th row of A237048, and of the sizes of the parts in SRS(n): 1 1 {1} {1} {1}
2 1 {1} {1} {3}
3 2 {1,0} {1,1} {2,2}
4 1 {1} {1,0} {7}
5 2 {1,0} {1,1} {3,3}
6 3 {1,1} {1,0,1} {12}
7 2 {1,0} {1,1,0} {4,4}
8 1 {1} {1,0,0} {15}
9 5 {1,0,1} {1,1,1} {5,3,5}
10 2 {1,0} {1,0,0,1} {9,9}
11 2 {1,0} {1,1,0,0} {6,6}
12 3 {1,1} {1,0,1,0} {28}
13 2 {1,0} {1,1,0,0} {7,7}
14 2 {1,0} {1,0,0,1} {12,12}
15 11 {1,0,1,1} {1,1,1,0,1} {8,8,8}
16 1 {1} {1,0,0,0,0} {31}
...
27 10 {1,0,1,0} {1,1,1,0,0,1} {14,6,6,14}
28 3 {1,1} {1,0,0,0,0,0,1} {56}
...
MATHEMATICA
(* function a237048[ ] is defined in A237048 *) b237048[n_] := Fold[2#1+Mod[#2, 2]&, 0, Flatten[Position[a237048[n], 1]]]
a352696[n_] := Map[b237048, Range[n]]
a352696[85]
CROSSREFS
Cf. A000975, A061854, A174905, A174973, A235791, A237048, A237270, A237591, A237593, A238443, A239929, A247687, A352030.
Primes p such that the largest Dyck path of the symmetric representation of sigma(p) has a central peak.
+10
1
2, 7, 17, 19, 29, 31, 47, 53, 67, 71, 73, 97, 101, 103, 127, 131, 157, 163, 167, 191, 193, 197, 199, 233, 239, 241, 251, 277, 281, 283, 293, 331, 337, 347, 349, 379, 383, 389, 397, 401, 439, 443, 449, 457, 461, 463, 499, 503, 509, 521, 523, 563, 569, 571, 577, 587, 593, 631, 641, 643, 647, 653, 659, 661
COMMENTS
Also primes p such that both Dyck paths of the symmetric representation of sigma(p) have a central peak.
Note that the symmetric representation of sigma of an odd prime consists of two perpendicular bars connected by an irregular zig-zag path (see example).
Odd primes and the terms of this sequence are easily identifiable in the pyramid described in A245092 (see Links section). For more information about the mentioned Dyck paths see A237593. Equivalently, primes p such that the largest Dyck path of the symmetric representation of sigma(p) has an odd number of peaks.
EXAMPLE
Illustration of initial terms:
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p sigma(p) Diagram of the symmetry of sigma
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7 8 |_ _ _ _| | | | |
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17 18 |_ _ _ _ _ _ _ _ _| |
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19 20 |_ _ _ _ _ _ _ _ _ _|
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For the first four terms of the sequence we can see in the above diagram that the largest Dyck path of the symmetric representation of sigma(p) has a central peak.
CROSSREFS
Cf. A000203, A065091, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A239660, A239929, A239931, A239933, A244050, A245092, A262626.
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