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A121860 a(n) = Sum_{d|n} n!/(d!*(n/d)!).

%I A121860 #28 Aug 12 2025 17:20:34
%S A121860 1,2,2,8,2,122,2,1682,10082,30242,2,7318082,2,17297282,3632428802,
%T A121860 36843206402,2,2981705126402,2,1690185726028802,3379030566912002,
%U A121860 28158588057602,2,76941821303636889602,1077167364120207360002
%N A121860 a(n) = Sum_{d|n} n!/(d!*(n/d)!).
%C A121860 a(n) = 2 iff n is prime.
%C A121860 a(468) has 1007 decimal digits. - _Michael De Vlieger_, Sep 12 2018
%C A121860 From _Gus Wiseman_, Jan 10 2019: (Start)
%C A121860 Number of matrices whose entries are 1,...,n, up to row and column permutations. For example, inequivalent representatives of the a(4) = 8 matrices are:
%C A121860   [1 2 3 4]
%C A121860 .
%C A121860   [1 2] [1 2] [1 3] [1 3] [1 4] [1 4]
%C A121860   [3 4] [4 3] [2 4] [4 2] [2 3] [3 2]
%C A121860 .
%C A121860   [1]
%C A121860   [2]
%C A121860   [3]
%C A121860   [4]
%C A121860 (End)
%C A121860 Conjecture: the sequence a(n) taken modulo a positive integer k >= 3 eventually becomes constant equal to 2. For example, the sequence taken modulo 11 is [1, 2, 2, 8, 2, 1, 2, 10, 6, 3, 2, 2, 2, 2, 2, 2, ...]. - _Peter Bala_, Aug 08 2025
%H A121860 Michael De Vlieger, <a href="/A121860/b121860.txt">Table of n, a(n) for n = 1..467</a>
%H A121860 Jimmy Devillet, Gergely Kiss, <a href="https://arxiv.org/abs/1806.02073">Characterizations of biselective operations</a>, arXiv:1806.02073 [math.RA], 2018.
%F A121860 E.g.f.: Sum_{k>0} (exp(x^k)-1)/k!.
%p A121860 with(numtheory): seq(n!*add(1/(d!*(n/d)!), d in divisors(n)), n = 1..25); # _Peter Bala_, Aug 04 2025
%t A121860 f[n_] := Block[{d = Divisors@n}, Plus @@ (n!/(d! (n/d)!))]; Array[f, 25] (* _Robert G. Wilson v_, Sep 11 2006 *)
%t A121860 Table[DivisorSum[n, n!/(#!*(n/#)!) &], {n, 25}] (* _Michael De Vlieger_, Sep 12 2018 *)
%o A121860 (PARI) a(n) = sumdiv(n, d, n!/(d!*(n/d)!)); \\ _Michel Marcus_, Sep 13 2018
%Y A121860 Cf. A038041, A057625, A061095, A100195, A236696, A258899, A258903, A320444, A386877.
%K A121860 easy,nonn
%O A121860 1,2
%A A121860 _Vladeta Jovovic_, Sep 09 2006
%E A121860 More terms from _Robert G. Wilson v_, Sep 11 2006