David Broadhurst - cp's OEIS frontend

A381289 Number of subsets of 6 integers between 1 and n such that their sum is 0 modulo n.

1, 3, 10, 20, 42, 76, 132, 212, 335, 497, 728, 1028, 1428, 1932, 2586, 3384, 4389, 5601, 7084, 8844, 10966, 13442, 16380, 19780, 23751, 28301, 33566, 39536, 46376, 54086, 62832, 72624, 83661, 95931, 109668, 124872, 141778, 160398, 181006

Offset: 7

Xavier Roulleau and David Broadhurst, Feb 19 2025

For an integer s multiple of 6, this is also the number of subsets of 6 integers between 1 and n such that their sum is s modulo n.

			For n=8, there are a(8)=3 order 6 subsets of Z/8Z with sum equal to 0 mod 8.

David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,4,-3,-3,4,1,-1,-3,1,2,-1).

Cf. A011796.

G.f.: x^7*(1 + x + 3*x^2 + 2*x^4 + 4*x^5 + x^6 - x^7 + x^8)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)*(1 - x^6)). - David Broadhurst, Feb 19 2025

A381291 Number of subsets of 8 integers between 1 and n such that their sum is 0 modulo n.

Original entry on oeis.org

1, 5, 15, 43, 99, 217, 429, 809, 1430, 2438, 3978, 6310, 9690, 14550, 21318, 30666, 43263, 60115, 82225, 111041, 148005, 195143, 254475, 328755, 420732, 534076, 672452, 840652, 1043460, 1287036, 1577532, 1922740, 2330445, 2810385, 3372291, 4028183, 4790071

Offset: 9

Xavier Roulleau and David Broadhurst, Feb 19 2025

nonn

For an integer s multiple of 8, this is also the number of subsets of 8 integers between 1 and n such that their sum is s modulo n.

			For n=10, there are a(10)=5 order 8 subsets of Z/10Z with sum equal to 0 mod 10.

David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025.
Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1).

Cf. A381290, A381289, A011796.

G.f.: x^9*(1 + x - x^2 + 7*x^3 - 4*x^4 + 6*x^5 + 4*x^6 - 4*x^7 + 3*x^8 + 5*x^9 - 3*x^10 + x^11)/((1 - x)^4*(1 - x^2)^2*(1 - x^4)*(1 - x^8)).

A381290 Number of subsets of 6 integers between 1 and n such that their sum is 1 modulo n.

Original entry on oeis.org

1, 4, 9, 22, 42, 78, 132, 217, 333, 504, 728, 1035, 1428, 1944, 2583, 3399, 4389, 5616, 7084, 8866, 10962, 13468, 16380, 19806, 23751, 28336, 33561, 39576, 46376, 54126, 62832, 72675, 83655, 95988, 109668, 124929, 141778, 160468, 180999

Offset: 7

Xavier Roulleau and David Broadhurst, Feb 19 2025

For s an integer such that GCD(s,6)=1, this is also the number of subsets of 6 integers between 1 and n such that their sum is s modulo n.

			For n=7, a(7)=1 since the set {0,1,2,3,4,5} is the unique order 6 subset of Z/7Z with sum equal to 1 mod 7.

David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,4,-3,-3,4,1,-1,-3,1,2,-1).

Cf. A381289, A011796.

G.f.: x^7*(1 + 2*x + 3*x^3 + 2*x^4 + 2*x^5 + x^6 + x^7)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)*(1 - x^6)).

A381219 a(n) = (6^n+2^n-23^n)(n-1)!/2.

Original entry on oeis.org

1, 11, 170, 3450, 87864, 2715720, 99248400, 4200210000, 202383054720, 10949741066880, 657619863264000, 43423960900320000, 3127284944109849600, 243957907264508236800, 20493712266753293568000, 1844490309401727187200000, 177073768932670444843008000, 18061662138488384327847936000, 1950666948832313303630438400000

Offset: 1

N. J. A. Sloane, Feb 17 2024, based on an email from David Broadhurst

nonn

This is the sequence associated with the gamma-chain (see Broadhurst link).

David Broadhurst, Resurgent Integer Sequences, Rutgers Experimental Math Seminar, Feb 06 2025; Slides.

Cf. A379809.

Table[(6^n+2^n-2*3^n) (n-1)!/2,{n,20}] (* Harvey P. Dale, Aug 13 2025 *)

D-finite with recurrence a(n) +11*(-n+1)*a(n-1) +36*(n-1)*(n-2)*a(n-2) -36*(n-1)*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Feb 18 2025

A352923 Let c(s) denote A109812(s). Suppose c(s) = 2^n - 1, and define m(n), p(n), r(n) by m(n) = c(s-1)/2^n, p(n) = c(s+1)/2^n, r(n) = max(m(n), p(n)); sequence gives r(n).

Original entry on oeis.org

1, 2, 4, 4, 6, 7, 8, 9, 10, 11, 12, 14, 14, 16, 18, 18, 18, 20

Offset: 1

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 24 2022)

The sequences m, p, r are well-defined since every number appears in A109812, and if A109812(s) = 2^n - 1, then by definition both A109812(s-1) and A109812(s+1) must be multiples of 2^n.
The sequences m, p, r are discussed in A352920.
Conjecture: r(n) >= n for n >= 1.

Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352922.

A352922 Let c(s) denote A109812(s). Suppose c(s) = 2^n - 1, and define m(n), p(n), r(n) by m(n) = c(s-1)/2^n, p(n) = c(s+1)/2^n, r(n) = max(m(n), p(n)); sequence gives m(n).

Original entry on oeis.org

0, 1, 4, 3, 6, 6, 8, 8, 10, 10, 11, 14, 14, 16, 18, 18, 18, 20

Offset: 1

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 24 2022)

The sequences m, p, r are well-defined since every number appears in A109812, and if A109812(s) = 2^n - 1, then by definition both A109812(s-1) and A109812(s+1) must be multiples of 2^n.
The sequences m, p, r are discussed in A352920.
(We assume A109812(0)=0 in order to get m(1)=0.)

Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352923.

A352921 Let c(s) denote A109812(s). Suppose c(s) = 2^n - 1, and define m(n), p(n), r(n) by m(n) = c(s-1)/2^n, p(n) = c(s+1)/2^n, r(n) = max(m(n), p(n)); sequence gives p(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 9, 9, 11, 12, 13, 13, 15, 15, 17, 17, 19

Offset: 1

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 24 2022)

The sequences m, p, r are well-defined since every number appears in A109812, and if A109812(s) = 2^n - 1, then by definition both A109812(s-1) and A109812(s+1) must be multiples of 2^n.

The sequences m, p, r are discussed in A352920.

Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352923.

A352920 Values of A109812(k) where k/A109812(k) reaches a new high point.

Original entry on oeis.org

1, 3, 6, 7, 31, 63, 127, 511, 4093, 4094, 4095, 16383, 32767, 262143

Offset: 1

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 23 2022)

The corresponding values of k are given in A352919.
This is a subset of A352336.
It is not necessary for a term of this sequence to be of the form 2^k - 1: there may be a zero close to the end of the binary expansion.
It appears that n/A109812(n) is unbounded. The reasoning behind this is as follows.
Consider terms A109812(k) that are the form 2^i - 1 (see the Examples section).
For such k, we necessarily have
a(k+1) = p(i)*2^i and a(k-1) = m(i)*2^i,
with integers p(i) and m(i). Let r(i) = max(p(i), m(i)).
Taking A109812(0) = 0, we have the following values:
  i : 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18
p(i): 1 2 3 4 5 7 7 9  9 11 12 13 13 15 15 17 17 19 [A352921]
m(i): 0 1 4 3 6 6 8 8 10 10 11 14 14 16 18 18 18 20 [A352922]
r(i): 1 2 4 4 6 7 8 9 10 11 12 14 14 16 18 18 18 20 [A352923]
for i < 19.  Furthermore, from the graphs in the entry A109812 it appears that r(19) = 21, r(20) = 22, r(21) = 22, r(22) = 24. The corresponding four values of k/aA109812(k) are, approximately, 6.42199, 6.80074, 6.88852, 7.39979.
This suggests the following conjecture:
Conjecture: r(k) > k for all k > 4.
Combining this with the conjecture that A109812(k)/k is bounded (see A352919 and A352920), we have:
Conjecture: k/A109812(k) is unbounded.

			Let c(k) denote A109812(k). The first 14 record high-points of k/c(k) are as follows:
[k/c(k), k, c(k), "binary(c(n))"]
[1.000000000 1 1 "1"]
[1.333333333 4 3 "11"]
[1.500000000 9 6 "110"]
[2.285714286 16 7 "111"]
[2.451612903 76 31 "11111"]
[2.571428571 162 63 "111111"]
[3.291338583 418 127 "1111111"]
[3.702544031 1892 511 "111111111"]
[4.665037870 19094 4093 "111111111101"]
[4.713727406 19298 4094 "111111111110"]
[4.898412698 20059 4095 "111111111111"]
[5.167124458 84653 16383 "11111111111111"]
[5.327494125 174566 32767 "111111111111111"]
[6.439611205 1688099 262143 "111111111111111111"]
The values of k and c(k) form A352919 and the present sequence.

Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352923.

A352919 Indices k where k/A109812(k) reaches a new high point.

Original entry on oeis.org

1, 4, 9, 16, 76, 162, 418, 1892, 19094, 19298, 20059, 84653, 174566, 1688099

Offset: 1

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 23 2022)

The corresponding values of A109812(k) are given in A352920.

This is a subset of A352359.

			Let c(k) denote A109812(k). The first 14 record high-points of k/c(k) are as follows:
[k/c(k), k, c(k), "binary(c(n))"]
[1.000000000 1 1 "1"]
[1.333333333 4 3 "11"]
[1.500000000 9 6 "110"]
[2.285714286 16 7 "111"]
[2.451612903 76 31 "11111"]
[2.571428571 162 63 "111111"]
[3.291338583 418 127 "1111111"]
[3.702544031 1892 511 "111111111"]
[4.665037870 19094 4093 "111111111101"]
[4.713727406 19298 4094 "111111111110"]
[4.898412698 20059 4095 "111111111111"]
[5.167124458 84653 16383 "11111111111111"]
[5.327494125 174566 32767 "111111111111111"]
[6.439611205 1688099 262143 "111111111111111111"]
The values of k and c(k) form the present sequence and A352920.

Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352923.

A352918 Values of A109812(k) where A109812(k)/k reaches a new high point.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 96, 128, 320, 512, 2048, 2304, 19922944, 41943040, 167772160

Offset: 1

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 21 2022)

The corresponding values of k are given in A352917.
This is a subset of A352203.
The slow growth of A109812(k)/k (see Examples section) suggests that A109812(k)/k is bounded. That is, it appears there is a constant c (between 3.7 and 4) such that A109812(k) < c*k for all k.

			Let c(k) denote A109812(k). The first 15 record high-points of c(k)/k are as follows:
[c(k)/k, k, c(k), "binary(c(n))"]
[1.000000000, 1, 1, "1"]
[1.333333333, 3, 4, "100"]
[1.600000000, 5, 8, "1000"]
[2.000000000, 8, 16, "10000"]
[2.133333333, 15, 32, "100000"]
[2.206896552, 29, 64, "1000000"]
[2.400000000, 40, 96, "1100000"]
[2.560000000, 50, 128, "10000000"]
[2.962962963, 108, 320, "101000000"]
[3.121951220, 164, 512, "1000000000"]
[3.155624037, 649, 2048, "100000000000"]
[3.539170507, 651, 2304, "100100000000"]
[3.616182275, 5509386, 19922944, "1001100000000000000000000"]
[3.721304271, 11271059, 41943040, "10100000000000000000000000"]
[3.727433952, 45010096, 167772160, "1010000000000000000000000000"]
The values of k and c(k) form A352917 and the present sequence.

Cf. A109812, A352203, A352204, A352917.

cp's OEIS Frontend

User: David Broadhurst

A381289 Number of subsets of 6 integers between 1 and n such that their sum is 0 modulo n.

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A381291 Number of subsets of 8 integers between 1 and n such that their sum is 0 modulo n.

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Author

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Examples

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Formula

A381290 Number of subsets of 6 integers between 1 and n such that their sum is 1 modulo n.

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Author

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Examples

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Formula

A381219 a(n) = (6^n+2^n-2*3^n)*(n-1)!/2.

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A352923 Let c(s) denote A109812(s). Suppose c(s) = 2^n - 1, and define m(n), p(n), r(n) by m(n) = c(s-1)/2^n, p(n) = c(s+1)/2^n, r(n) = max(m(n), p(n)); sequence gives r(n).

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A352922 Let c(s) denote A109812(s). Suppose c(s) = 2^n - 1, and define m(n), p(n), r(n) by m(n) = c(s-1)/2^n, p(n) = c(s+1)/2^n, r(n) = max(m(n), p(n)); sequence gives m(n).

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A352921 Let c(s) denote A109812(s). Suppose c(s) = 2^n - 1, and define m(n), p(n), r(n) by m(n) = c(s-1)/2^n, p(n) = c(s+1)/2^n, r(n) = max(m(n), p(n)); sequence gives p(n).

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A352920 Values of A109812(k) where k/A109812(k) reaches a new high point.

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A352919 Indices k where k/A109812(k) reaches a new high point.

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A352918 Values of A109812(k) where A109812(k)/k reaches a new high point.

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A381219 a(n) = (6^n+2^n-23^n)(n-1)!/2.