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A320948 Array T(n,k) of number of Schur rings over Z_{p^n} where n>=1 for p odd and k-th prime (by descending antidiagonals). 0
2, 3, 7, 4, 13, 25, 4, 21, 58, 92, 6, 21, 113, 263, 345, 5, 43, 113, 614, 1203, 1311, 6, 31, 313, 614, 3351, 5531, 5030, 4, 43, 196, 2288, 3351, 18329, 25511, 19439, 6, 21, 313, 1247, 16749, 18329, 100372, 117910, 75545, 8, 43, 113, 2288, 7953, 122675, 100372, 550009, 545730, 294888 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..55.

Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015 [table 5.1 p. 22].

EXAMPLE

Array begins:

k:    2      3       4       5       6       7       8

n\p   3      5       7      11      13      17      19

1     2,     3,      4,      4,      6,      5,      6, ...

2     7,    13,     21,     21,     43,     31,     43, ...

3    25,    58,    113,    113,    313,    196,    313, ...

4    92,   263,    614,    614,   2288,   1247,   2288, ...

5   345,  1203,   3351,   3351,  16749,   7953,  16749, ...

6  1311,  5531,  18329,  18329, 122675,  50775, 122675, ...

7  5030, 25511, 100372, 100372, 898706, 324323, 898706, ...

...

MATHEMATICA

rows = 10;

om[n_] := om[n] = x om[n-1] + Sum[(CatalanNumber[k-1] x + 1) om[n - k], {k, 2, n}] // Expand; om[0] = 1; om[1] = x;

T = Table[Array[om, rows] /. x -> x0, {x0, DivisorSigma[0, #-1]& /@ Prime[ Range[2, rows+1]]}] // Transpose;

Table[T[[n-k+2, k-1]], {n, 1, rows}, {k, n+1, 2, -1}] // Flatten

CROSSREFS

Cf. A269750, A270785, A270786, A270787.

Sequence in context: A287950 A287628 A319863 * A086885 A324598 A229794

Adjacent sequences:  A320945 A320946 A320947 * A320949 A320950 A320951

KEYWORD

nonn,tabl

AUTHOR

Jean-Fran├žois Alcover, Oct 24 2018

STATUS

approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)