A334370
Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k) / prime(k)!).
Original entry on oeis.org
1, 0, 1, 1, 0, 11, 0, 22, 56, 36, 2640, 1, 8712, 79, 72436, 360465, 48608, 49008961, 794376, 4232764, 7753140, 942565890, 18198334, 14799637777, 10577976, 366619314900, 2785137222400, 1475339135400, 1065920156634060, 3765722000041, 5869315258699050
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
(p-> `if`(p>n, 0, b(n-p, i-1)*binomial(n, p)))(ithprime(i))))
end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..30); # Alois P. Heinz, Jul 18 2023
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nmax = 30; CoefficientList[Series[Product[(1 + x^Prime[k]/Prime[k]!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[DivisorSum[k, -#/(-#!)^(k/#) &, PrimeQ[#] &] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 30}]
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+isprime(k)*x^k/k!))) \\ Seiichi Manyama, Feb 27 2022
A364327
Number of endofunctions on [n] such that the number of elements that are mapped to i is either 0 or a divisor of i.
Original entry on oeis.org
1, 1, 3, 13, 115, 851, 13431, 144516, 2782571, 47046307, 1107742273, 19263747713, 657152726011, 13657313316986, 451605697223110, 13377063396461138, 531234399267707419, 14563460779785318719, 721703507708044677945, 22141894282020163910406, 1123287408943765640907425
Offset: 0
a(0) = 1: ().
a(1) = 1: (1).
a(2) = 3: (22), (21), (12).
a(3) = 13: (333), (322), (232), (223), (321), (231), (213), (312), (132), (123), (221), (212), (122).
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+add(
`if`(d>n, 0, b(n-d, i-1)*binomial(n, d)), d=numtheory[divisors](i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23);
A364328
Number of endofunctions on [n] such that the number of elements that are mapped to i is either 0 or a prime divisor of i.
Original entry on oeis.org
1, 0, 1, 1, 6, 21, 110, 904, 4312, 74400, 731412, 5600761, 128196024, 792051157, 18696610816, 264267572121, 7136433698464, 57948743342529, 2228312959187256, 22463157401776612, 681974906329502904, 15395459281239915282, 463374873030990445252, 6091833036158810701465
Offset: 0
a(0) = 1: ().
a(2) = 1: (22).
a(3) = 1: (333).
a(4) = 6: (4422), (4242), (4224), (2442), (2424), (2244).
a(5) = 21: (55555), (44333), (43433), (43343), (43334), (34433), (34343), (34334), (33443), (33434), (33344), (33322), (33232), (33223), (32332), (32323), (32233), (23332), (23323), (23233), (22333).
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+add(
`if`(d>n, 0, b(n-d, i-1)*binomial(n, d)), d=numtheory[factorset](i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23);
Showing 1-3 of 3 results.
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