A003964 Fully multiplicative with a(prime(k)) = partition(k+1).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 14, 15, 16, 22, 18, 30, 20, 21, 22, 42, 24, 25, 30, 27, 28, 56, 30, 77, 32, 33, 44, 35, 36, 101, 60, 45, 40, 135, 42, 176, 44, 45, 84, 231, 48, 49, 50, 66, 60, 297, 54, 55, 56, 90, 112, 385, 60, 490, 154, 63, 64, 75, 66, 627, 88, 126, 70

Offset: 1

Author

Marc LeBrun, N. J. A. Sloane

Keywords

• nonn
• look
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000041, A000720, A299200.

Programs

Maple

with(numtheory): with(combinat):
a:= n-> mul(numbpart(pi(i[1])+1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..82);  # Alois P. Heinz, Jan 14 2021

Mathematica

f[p_, e_] := PartitionsP[PrimePi[p] + 1]^e; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

PARI

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = numbpart(primepi(f[k, 1])+1)); factorback(f); \\ Michel Marcus, Jan 14 2021

Formula

If n = Product p(k)^e(k) then a(n) = Product partition(k+1)^e(k).

Multiplicative with a(p^e) = A000041(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001

Sum_{n>=1} 1/a(n) = 1 / Product_{k>=2} (1 - 1/A000041(k)) = 6.16770060042144081793... . - Amiram Eldar, Sep 19 2023

Extensions

Description corrected and sequence extended by David W. Wilson