Joseph E. Marrow - cp's OEIS frontend

A384156 Number of group Schur rings of the cyclic group Z_n.

Original entry on oeis.org

1, 1, 2, 3, 3, 7, 4, 10, 7, 10, 34, 32, 6, 13, 21

Offset: 1

Joseph E. Marrow, May 20 2025

Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015.
Andrew Misseldine, Enumeration Techniques on Cyclic Schur Rings, arXiv preprint arXiv:2112.03422 [math.GR], 2021.
Matan Ziv-Av, Enumeration of Schur Rings Over Small Groups, Lecture Notes in Computer Science, International Workshop on Computer Algebra in Scientific Computing, Sept 2014. DOI:10.1007/978-3-319-10515-4_35

Cf. A270785, A270786, A270787, A270789, A112951.

A383219 Number of nilpotent semigroups by order, up to isomorphism and anti-isomorphism.

Original entry on oeis.org

0, 0, 1, 2, 10, 93, 2813, 616830, 1833587417, 52972875977730

Offset: 0

Joseph E. Marrow, Apr 19 2025

Andreas Distler, Classification and Enumeration of Finite Semigroups, PhD Thesis, University of St. Andrews, (2010).

Cf. A001423.

A331096 Numbers k such that the sum of all divisors except k, minus the sum of the digits of k, is equal to k.

Original entry on oeis.org

20, 66, 138, 174, 246, 282, 318, 354, 426, 534, 606, 642, 822, 1038, 1074, 1146, 1182, 1362, 1434, 1506, 1542, 1614, 1902, 2082, 2118, 2154, 2334, 2406, 2514, 2802, 3018, 3054, 3126, 3342, 3414, 3522, 3702, 4062, 4206, 4314, 5034, 5142, 5322, 6114, 7122, 7232, 7302, 8202

Offset: 1

Joseph E. Marrow, Jan 08 2020

The first two odd elements are a(49) = 8415 and a(107) = 31815.- Robert Israel, Jan 16 2020

Robert Israel, Table of n, a(n) for n = 1..1400

Cf. A007953 (sum of digits), A001065 (sum of proper divisors).

Related sequences are A331037 and A331093.

[k:k in [1..8250]| DivisorSigma(1,k) eq 2*k+&+Intseq(k)]; // Marius A. Burtea, Jan 11 2020

filter:= proc(k) numtheory:-sigma(k)-convert(convert(k,base,10),`+`)=2*k end proc:
select(filter, [$1..10000]); # Robert Israel, Jan 16 2020

Select[Range[10^4], DivisorSigma[1, #] - Plus @@ IntegerDigits[#] == 2 # &] (* Amiram Eldar, Jan 11 2020 *)

isok(k) = sigma(k) - k - sumdigits(k) == k; \\ Michel Marcus, Jan 11 2020

A331093 Numbers such that the sum of their divisors, excluding 1 and the number itself, minus the sum of their digits equals the number.

Original entry on oeis.org

12, 114256, 6988996, 8499988, 8689996, 8789788, 8877988, 8988868, 8999956, 9696988, 9759988, 9899596, 9948988, 9996868, 9998884, 9999892, 15996988, 16878988, 17799796, 17887996, 17988796, 17999884, 18579988, 18768988, 18869788, 18895996, 18958996, 18995788, 19398988, 19587988, 19698868, 19777996, 19799668

Offset: 1

Joseph E. Marrow, Jan 08 2020

After the second term, it seems that the digit sum is 55.
All terms after a(2) appear to be of the form 2^2 * 7 * p, where p is a prime. - Scott R. Shannon, Jan 09 2020
If there exists a third term not of the form 2^2*7*p, it is larger than 10^13. - Giovanni Resta, Jan 14 2020

			a(3) = 6988996 as the sum of the divisors of 6988996, excluding 1 and 6988996, equals 6989051, the sum of its digits equals 55, and 6989051 - 55 = 6988996.

Cf. A000203, A007953, A048050.

Cf. A331037 (sum of divisors + digit sum = number).

Select[Range[10^7], DivisorSigma[1, #] - Plus @@ IntegerDigits[#] == 2 # + 1 &] (* Amiram Eldar, Jan 08 2020 *)

isok(n) = sigma(n) - n - 1 - sumdigits(n) == n; \\ Michel Marcus, Jan 09 2020

Terms a(7) and beyond from Scott R. Shannon, Jan 09 2020

A331037 Numbers k such that the sum of the divisors of k (except for 1 and k) plus the sum of the digits of k is equal to k.

Original entry on oeis.org

1, 2, 3, 5, 7, 14, 52, 76, 2528, 9536, 9664, 35456, 138496, 8456192, 33665024, 33673216, 537444352, 2148958208, 137454419968

Offset: 1

Joseph E. Marrow, Jan 08 2020

Additional terms include 537444352, 2148958208, 137454419968, 35184644718592, 9007202811510784. Are there any terms > 1 not of the form 2^k*p where p is prime and k>0? - David A. Corneth, Jan 08 2020
Terms not of the form 2^k*p do exist, for example 2^15*65713*24194197 and 2^19*1739719*2639431. - Giovanni Resta, Jan 08 2020
a(20) > 10^13. - Giovanni Resta, Jan 14 2020

			The first term that is not 1 or a single-digit prime is obtained by adding the proper divisors of 14 other than 1 (2,7) to its digits (1,4): (2+7) + (1+4) = 14.
The second such term is 52: the proper divisors of 52 other than 1 (2,4,13,26) and its digits (5,2) sum to (2+4+13+26) + (5+2) = 52.

Cf. A000203, A007953, A048050.

Cf. A331093 (sum of divisors - digit sum = the number).

Select[Range[10^7], DivisorSigma[1, #] - # - If[# == 1, 0, 1] + Plus @@ IntegerDigits[#] == # &] (* Amiram Eldar, Jan 12 2020 *)

is(n) = n == sigma(n)-1-if(n>1,n,0)+sumdigits(n) \\ Rémy Sigrist, Jan 08 2020

a(14)-a(16) from Rémy Sigrist, Jan 08 2020

a(17)-a(19) from Giovanni Resta, Jan 14 2020

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User: Joseph E. Marrow

A384156 Number of group Schur rings of the cyclic group Z_n.

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A383219 Number of nilpotent semigroups by order, up to isomorphism and anti-isomorphism.

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A331096 Numbers k such that the sum of all divisors except k, minus the sum of the digits of k, is equal to k.

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A331093 Numbers such that the sum of their divisors, excluding 1 and the number itself, minus the sum of their digits equals the number.

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A331037 Numbers k such that the sum of the divisors of k (except for 1 and k) plus the sum of the digits of k is equal to k.

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Mathematica

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