Displaying 221-230 of 271 results found.
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Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n), multiplied by n.
+10
1
1, 6, 6, 6, 28, 15, 15, 72, 28, 28, 120, 45, 27, 45, 90, 90, 66, 66, 336, 91, 91, 168, 168, 120, 120, 120, 496, 153, 153, 702, 190, 190, 840, 231, 105, 105, 231, 396, 396, 276, 276, 1440, 325, 125, 325, 546, 546, 378, 162, 162, 378, 1568, 435, 435, 2160, 496, 496, 2016
COMMENTS
Since both A000203(n) and A024916(n) have a symmetric representation then both row n and the triangle have can be represented as a symmetric polycube.
EXAMPLE
The irregular triangle begins:
1;
6;
6, 6;
28;
15, 15;
72;
28, 28;
120;
45, 27, 45;
90, 90;
66, 66;
336;
91, 91;
168, 168;
120, 120, 120;
496;
153, 153;
702;
190, 190;
840;
231, 105, 105, 231;
...
For n = 9 the parts of the symmetric representation of sigma(9) are [5, 3, 5], so row 9 is [45, 27, 45].
CROSSREFS
Cf. A000203, A024916, A064987, A143128, A185283, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A244580.
Numbers n such that in the symmetric representation of sigma(n) all parts are of the same size.
+10
1
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 52, 53, 54, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 71, 72, 73, 74, 76, 78, 79, 80, 82, 83, 84, 86, 88, 89, 90, 92, 94, 96, 97, 100
COMMENTS
All powers of 2, all prime numbers and all even perfect numbers are members of this sequence.
For more information about the symmetric representation of sigma see A237270 and A237593.
EXAMPLE
9 is not in the sequence because the parts of the symmetric representation of sigma(9) = 13 are [5, 3, 5].
10 is in the sequence because the parts of the symmetric representation of sigma(10) = 18 are [9, 9].
CROSSREFS
Cf. A000040, A000079, A000203, A000396, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931- A239934, A245092, A262626, A266000.
Numbers n such that the symmetric representation of sigma(n) has at least two parts of distinct size.
+10
1
9, 21, 25, 27, 33, 35, 39, 45, 49, 50, 51, 55, 57, 63, 65, 69, 70, 75, 77, 81, 85, 87, 91, 93, 95, 98, 99, 105, 110, 111, 115, 117, 119, 121, 123, 125
COMMENTS
In other words: numbers n such that the symmetric representation of sigma(n) has at least two parts with distinct number of cells.
For more information about the symmetric representation of sigma see A237270 and A237593.
EXAMPLE
The symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.
._ _ _ _ _ 5
|_ _ _ _ _|
. |_ _ 3
. |_ |
. |_|_ _ 5
. | |
. | |
. | |
. | |
. . . . . . . . |_| . . x
.
There are three parts: 5 + 3 + 5 = 13, so 9 is in the sequence because the structure contains at least two parts of distinct size.
CROSSREFS
Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931- A239934, A245092, A262626.
Irregular triangle read by rows in which row n lists the semiperimeters of the parts of the symmetric representation of sigma(n).
+10
1
2, 4, 3, 3, 8, 4, 4, 12, 5, 5, 16, 6, 4, 6, 10, 10, 7, 7, 24, 8, 8, 13, 13, 9, 8, 9, 32
EXAMPLE
Triangle begins:
2;
4;
3, 3;
8;
4, 4;
12;
5, 5;
16;
6, 4, 6;
10, 10;
7, 7;
24;
8, 8;
13, 13;
9, 8, 9;
32;
...
For n = 9 the symmetric representation of sigma(9) = 13 has three parts of areas 5, 3, 5 respectively. The perímeters of the parts are 12, 8 and 12, so the 9th row of triangle lists the semiperimeters: 6, 4, 6.
Irregular triangle read by rows in which the n-th row lists the number of legs in the parts of the symmetric representation of sigma(n).
+10
1
2, 2, 1, 1, 4, 1, 1, 6, 1, 1, 6, 1, 2, 1, 3, 3, 1, 1, 8, 1, 1, 3, 3, 1, 6, 1, 10, 1, 1, 10, 1, 1, 10, 1, 3, 3, 1, 3, 3, 1, 1, 12, 1, 4, 1, 3, 3, 1, 3, 3, 1, 14, 1, 1, 14, 1, 1, 14, 1, 3, 3, 1, 3, 3, 1, 6, 1, 16, 1, 1, 3, 3, 1, 3, 3, 1, 16, 1, 1, 16, 1, 1, 7, 7, 1, 14, 1, 3, 3, 1, 1, 18, 1, 6, 1, 3, 10, 3, 1, 3, 3, 1, 7, 7
COMMENTS
The legs are those line segments in the parts of the symmetric representation of sigma(n) that bound a portion of its nonzero area.
Blocks of nonzero numbers start and end at odd positions in the rows of triangle A249223 unless a block extends to the end of the row. Therefore, the number of legs in any part of the symmetric representation of sigma(n) is odd when A237271(n) is even, and odd except for the middle part when A237271(n) is odd.
EXAMPLE
Since row 14 of triangle A249223 is 1 1 1 0 the symmetric representation of sigma(14) has two parts of three legs each and row 14 in this triangle is 3 3. Since row 15 of triangle A249223 is 1 0 1 1 2 the symmetric representation of sigma(15) has three parts of 1 leg, 6 legs, and 1 leg, respectively, and row 15 in this triangle is 1 6 1. Irregular triangle of legs of parts:
1: 2
2: 2
3: 1 1
4: 4
5: 1 1
6: 6
7: 1 1
8: 6
9: 1 2 1
10: 3 3
11: 1 1
12: 8
13: 1 1
14: 3 3
15: 1 6 1
16: 10
17: 1 1
18: 10
19: 1 1
20: 10
21: 1 3 3 1
...
Illustration of the legs for the symmetric representations of sigma(1)..sigma(24); for comparison see also A237593. The legs of the central parts of the symmetric representation of sigma for 9, 15 and 21 have 3, 3 and 4 parts, and touch the legs of 8, 14 and 20, respectively. . _ _ _ _ _ _ _ _ _ _ _ _ _
. _ _ _ _ _ _ _ _ _ _ _ _ |
. _ _ _ _ _ _ _ _ _ _ _ _ |
. _ _ _ _ _ _ _ _ _ _ _ | |
. _ _ _ _ _ _ _ _ _ _ _ | |_ _ _
. _ _ _ _ _ _ _ _ _ _ | | |
. _ _ _ _ _ _ _ _ _ _ | |_ _ |_
. _ _ _ _ _ _ _ _ _ | |_ _ _ |_
. _ _ _ _ _ _ _ _ _ | |_ _ |_ |_ _
. _ _ _ _ _ _ _ _ | |_ _ |_ _ |
. _ _ _ _ _ _ _ _ | | | |_ |
. _ _ _ _ _ _ _ | |_ _ |_ |_ | | |_ _ _ _
. _ _ _ _ _ _ _ |_ _ |_ |_ _ | | |_ _ _ _ |
. _ _ _ _ _ _ | |_ |_ |_ | |_|_ _ _ | | |
. _ _ _ _ _ _ |_ _ |_ | |_ _ _ | | | | |
. _ _ _ _ _ | |_ | |_ _ _ | | | | | | |
. _ _ _ _ _ |_ | |_|_ _ | | | | | | | | |
. _ _ _ _ |_ _ |_ _ | | | | | | | | | | |
. _ _ _ _ |_ | |_ _ | | | | | | | | | | | | |
. _ _ _ |_ |_|_ | | | | | | | | | | | | | | |
. _ _ _ |_ | | | | | | | | | | | | | | | | |
. _ _ |_ | | | | | | | | | | | | | | | | | | |
. _ _ | | | | | | | | | | | | | | | | | | | | |
. _ | | | | | | | | | | | | | | | | | | | | | | |
. | | | | | | | | | | | | | | | | | | | | | | | |
.
n: 1 2 3 4 5 6 7 8..10..12..14..16..18..20..22..24
.
MATHEMATICA
a279104[n_] := Map[Length, Select[SplitBy[a262045[n], #!=0&], First[#]!=0&]]
Flatten[Map[a279104, Range[52]]] (* sequence data for 52 rows *)
Irregular triangle read by rows in which row n lists the perimeters of the parts of the symmetric representation of sigma(n).
+10
1
4, 8, 6, 6, 16, 8, 8, 24, 10, 10, 32, 12, 8, 12, 20, 20, 14, 14, 48, 16, 16, 26, 26, 18, 16, 18, 64
EXAMPLE
4;
8;
6, 6;
16;
8, 8;
24;
10, 10;
32;
12, 8, 12;
20, 20;
14, 14;
48;
16, 16;
26, 26;
18, 16, 18;
64;
...
Illustration of the 9th row:
. 12
. _ _ _ _ _
. |_ _ _ _ _|
. _ _ 8
. |_ |
. |_| _
. | |
. | |
. | | 12
. | |
. |_|
.
For n = 9 the symmetric representation of sigma(9) = 13 has three parts of areas 5, 3, 5 respectively. The perímeters of the parts are 12, 8 and 12 as shown above, so the 9th row of triangle is 12, 8, 12.
1, 3, 2, 5, 2, 8, 2, 9, 5, 10, 2, 18, 2, 12, 10, 17, 2, 23, 2, 24, 12, 16, 2, 38, 7, 18, 14, 30, 2, 44, 2, 33, 16, 22, 14, 57, 2, 24, 18, 52, 2, 56, 2, 42, 34, 28, 2, 78, 9, 45, 22, 48, 2, 68, 18, 66, 24, 34, 2, 110, 2, 36, 42, 65, 20, 80, 2, 60, 28, 76, 2, 125, 2, 42, 50, 66, 20, 92, 2, 108, 41, 46, 2, 142, 24, 48, 34, 94, 2
COMMENTS
a(n) = 2 iff n is an odd prime ( A065091). Has a symmetric representation as a narrow pyramid with holes, in the same way as A249351.
EXAMPLE
. 1 - 0 = 1
. 3 - 0 = 3
. 4 - 2 = 2
. 7 - 2 = 5
. 6 - 4 = 2
. 12 - 4 = 8
...
CROSSREFS
Cf. A000203, A004526, A048050, A052928, A065091, A176059, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A249351, A262626, A279387, A281010.
a(n) is the sum of all divisors of all numbers k whose associated largest Dyck path contains the point (n,n) in the diagram of the symmetric representation of sigma(k) described in A237593, or 0 if no such k exists.
+10
1
1, 7, 13, 0, 20, 15, 43, 0, 66, 0, 24, 49, 59, 0, 134, 0, 60, 113, 0, 86, 0, 104, 165, 0, 48, 245, 0, 132, 0, 224, 0, 198, 0, 124, 57, 317, 0, 192, 0, 350, 0, 326, 0, 104, 211, 0, 434, 0, 216, 0, 0, 647, 0, 344, 0, 186, 331, 0, 584, 0, 270, 0, 234, 0, 672, 0, 350, 171, 0, 156, 639, 0, 672, 0, 390, 0, 368, 0, 956
CROSSREFS
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A259179, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.
Numbers m such that in the diagram of the symmetric representation of sigma(k) described in A237593 there is no Dyck path that contains the point (m,m), where both k and m are positive integers.
+10
1
4, 8, 10, 14, 16, 19, 21, 24, 27, 29, 31, 33, 37, 39, 41, 43, 46, 48, 50, 51, 53, 55, 58, 60, 62, 64, 66, 69, 72, 74, 76, 78, 80, 82, 83, 84, 87, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 114, 116, 119, 121, 123, 124, 125, 127, 129, 131, 133, 135, 138, 141, 143, 145, 147, 149, 151, 153
COMMENTS
Indices of the rows that contain a zero in the triangle A279385.
MATHEMATICA
a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a299482[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; Flatten[Position[t, 0]]]
Irregular triangle read by rows in which T(n,k) is the number of cells in the k-th level of the diagram of the symmetric representation of sigma(n).
+10
1
1, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 4, 3, 2, 2, 2, 2, 2, 2, 4, 5, 2, 2, 2, 2, 4, 3, 2, 2, 2, 4, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 6, 7, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 4, 2, 2, 2, 2, 2, 3, 6, 5, 2, 2, 2, 2, 2, 2, 4, 8, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 8, 9, 6, 2, 2, 2, 2, 2
COMMENTS
If n is an odd prime p then row n has length (p + 1)/2 and all terms in row n are 2's.
For more information about the diagram of the symmetric representation of sigma(n) see A237593 and other related sequences.
EXAMPLE
Triangle begins:
1;
2, 1;
2, 2;
2, 3, 2;
2, 2, 2;
2, 3, 4, 3;
2, 2, 2, 2;
2, 2, 4, 5, 2;
2, 2, 2, 4, 3;
2, 2, 2, 4, 6, 2;
2, 2, 2, 2, 2, 2;
2, 2, 2, 5, 6, 7, 4;
2, 2, 2, 2, 2, 2, 2;
2, 2, 2, 2, 4, 6, 4, 2;
2, 2, 2, 2, 3, 6, 5, 2;
2, 2, 2, 2, 2, 4, 8, 7, 2;
2, 2, 2, 2, 2, 2, 2, 2, 2;
2, 2, 2, 2, 2, 4, 8, 9, 6, 2;
2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...
CROSSREFS
Cf. A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A249351, A262611, A262626, A281010, A296508.
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