A082641 -id:A082641 - cp's OEIS frontend

A002956 Number of basic invariants for cyclic group of order and degree n.

1, 2, 4, 7, 15, 20, 48, 65, 119, 166, 348, 367, 827, 974, 1494, 2135, 3913, 4038, 7936, 8247, 12967, 17476, 29162, 28065, 49609, 59358, 83420, 97243, 164967, 152548, 280352, 295291, 405919, 508162, 674630, 708819, 1230259, 1325732, 1709230

Offset: 1

N. J. A. Sloane

a(n) is also the number of multisets of integers ranging from 1 to n, such that the sum of the members of the multiset is congruent to 0 mod n, and no submultiset exists whose sum of members is congruent to 0 mod n. These multisets can be thought of as partitions of n in modular arithmetic, thus this sequence can be thought of as a modular arithmetic version of the partition numbers (cf. A000041). - Andrew Weimholt, Jan 31 2011

M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 208.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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, to appear in Communications in Algebra.
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
Vadim Ponomarenko, Programs

Row sums of A082641.

Cf. A096337.

a(n) = A096337(n) + 1. - Filip Zaludek, Oct 26 2016

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.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A002956 Number of basic invariants for 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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A181887 a(0) = 0, and for n > 0, a(n) = A002956(n) - A000041(n).

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