A002956 -id:A002956 - cp's OEIS frontend

A181887 a(0) = 0, and for n > 0, a(n) = A002956(n) - A000041(n).

0, 0, 0, 1, 2, 8, 9, 33, 43, 89, 124, 292, 290, 726, 839, 1318, 1904, 3616, 3653, 7446, 7620, 12175, 16474, 27907, 26490, 47651, 56922, 80410, 93525, 160402, 146944, 273510, 286942, 395776, 495852, 659747, 690842

Offset: 0

Andrew Weimholt, Feb 01 2011

nonn

A002956 can be thought of as a modular arithmetic version of the partition numbers (A000041). The number of "modulo n" partitions of n is the number of multisets of integers ranging from 1 to n, such that the sum of members of the multiset is congruent to 0 mod n, and no submultiset exists whose members sum to 0 mod n. Therefore, a(n) is the number of "modulo n" partitions which are not ordinary partitions of n.

			The multisets counted by A002956(5) but not by A000041(5) are
..{1,3,3,3}
..{2,2,2,2,2}
..{2,2,2,4}
..{2,4,4}
..{3,3,3,3,3}
..{3,4,4,4}
..{3,3,4}
..{4,4,4,4,4}
So a(5) = 8.

Finklea, Moore, Ponomarenko and Turner, Invariant Polynomials and Minimal Zero Sequences

Cf. A000041, A002956, A082641

A082641 Triangle T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = number of basic invariants of degree k for the cyclic group of order and degree n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 4, 1, 3, 6, 6, 2, 2, 1, 3, 8, 12, 12, 6, 6, 1, 4, 10, 18, 16, 8, 4, 4, 1, 4, 14, 26, 32, 18, 12, 6, 6, 1, 5, 16, 36, 48, 32, 12, 8, 4, 4, 1, 5, 20, 50, 82, 70, 50, 30, 20, 10, 10, 1, 6, 24, 64, 104, 84, 36, 20, 12, 8, 4, 4, 1, 6, 28, 84, 168, 180, 132, 84, 60, 36, 24, 12, 12, 1, 7, 32, 104, 216, 242, 162, 96, 42, 30, 18, 12, 6, 6

Offset: 1

N. J. A. Sloane, May 15 2003

T(n,k) is also the number of multisets of k integers ranging from 1 to n, such that the sum of members of the multiset is congruent to 0 mod n, and no submultiset exists whose sum of members is congruent to 0 mod n. - Andrew Weimholt, Jan 31 2011

			Triangle with row sums (A002956):
  Z_1:  1  ................................... 1
  Z_2:  1  1  ................................ 2
  Z_3:  1  1  2  ............................. 4
  Z_4:  1  2  2  2  .......................... 7
  Z_5:  1  2  4  4  4  ...................... 15
  Z_6:  1  3  6  6  2  2  ................... 20
  Z_7:  1  3  8 12 12  6  6  ................ 48
  Z_8:  1  4 10 18 16  8  4  4  ............. 65
  Z_9:  1  4 14 26 32 18 12  6  6  ......... 119
  Z_10: 1  5 16 36 48 32 12  8  4  4  ...... 166
  Z_11: 1  5 20 50 82 70 50 30 20 10 10  ... 348
  ...

M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 208.
C. W. Strom, Complete systems of invariants of the cyclic groups of equal order and degree, Proc. Iowa Acad. Sci., 55 (1948), 287-290.

Finklea, Moore, Ponomarenko and Turner, Invariant Polynomials and Minimal Zero Sequences, Involve 1 (2008), no. 2, 159-165.
Bryson W. Finklea, Terri Moore, Vadim Ponomarenko and Zachary J. Turner, Invariant polynomials and minimal zero sequences, Involve, 1:2 (2008), pp. 159-165.
Vadim Ponomarenko, Table (Excel spread-sheet format)
Vadim Ponomarenko, Programs

Row sums give A002956.

More terms from Vadim Ponomarenko (vadim123(AT)gmail.com), Jun 29 2004

A238100 Number of canonical generators for Z_2 + Z_n.

Original entry on oeis.org

5, 20, 39, 166, 253, 974

Offset: 2

N. J. A. Sloane, Feb 25 2014

Bryson W. Finklea, Terri Moore, Vadim Ponomarenko and Zachary J. Turner, Invariant polynomials and minimal zero sequences, Involve, 1:2 (2008), pp. 159-165.

Cf. A002956, A082641, A238101, A238102.

A238101 Number of canonical generators for Z_3 + Z_n.

Original entry on oeis.org

20, 69, 367, 1494, 2642, 12967

Offset: 2

N. J. A. Sloane, Feb 25 2014

Bryson W. Finklea, Terri Moore, Vadim Ponomarenko and Zachary J. Turner, Invariant polynomials and minimal zero sequences, Involve, 1:2 (2008), pp. 159-165.

Cf. A002956, A082641, A238100, A238102.

A238102 Number of canonical generators for Z_4 + Z_n.

Original entry on oeis.org

39, 367, 1107, 8247, 19463, 97243

Offset: 2

N. J. A. Sloane, Feb 25 2014

Bryson W. Finklea, Terri Moore, Vadim Ponomarenko and Zachary J. Turner, Invariant polynomials and minimal zero sequences, Involve, 1:2 (2008), pp. 159-165.

Cf. A002956, A082641, A238101, A238100.

cp's OEIS Frontend

A181887 a(0) = 0, and for n > 0, a(n) = A002956(n) - A000041(n).

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A082641 Triangle T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = number of basic invariants of degree k for the cyclic group of order and degree n.

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A238100 Number of canonical generators for Z_2 + Z_n.

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A238101 Number of canonical generators for Z_3 + Z_n.

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A238102 Number of canonical generators for Z_4 + Z_n.

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