A072628 Number of divisors d of n such that d-1 is not prime.

1, 2, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 3, 4, 2, 2, 3, 4, 3, 4, 2, 5, 2, 3, 3, 4, 4, 4, 2, 3, 3, 5, 2, 4, 2, 4, 5, 4, 2, 3, 3, 6, 3, 5, 2, 4, 4, 5, 3, 4, 2, 5, 2, 3, 5, 4, 4, 6, 2, 4, 3, 7, 2, 4, 2, 3, 5, 4, 4, 6, 2, 6, 4, 4, 2, 5, 4, 4, 3, 5, 2, 7, 4, 5, 3, 4, 4, 4, 2, 4, 5, 7, 2, 5, 2, 5, 7

Offset: 1

Author

Labos Elemer, Jun 28 2002

Keywords

• nonn

Examples

			If n = p is prime then divisors - 1 = {1, p} - 1 = {0, p-1} so a(p) = 2 if p <> 3.
240 has 20 divisors, of them 8 divisors d have nonprime value of d-1, {0, 1, 4, 9, 14, 15, 39, 119}, so a(240) = 8.
		

Links

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

Crossrefs

Cf. A000005, A072627.

Programs

Mathematica

di[x_] := Divisors[x]; dp[x_] := Part[di[x], Flatten[Position[PrimeQ[ -1+di[x]], True]]]-1; Table[DivisorSigma[0, w]-Length[dp[w]], {w, 1, 128}]
a[n_] := DivisorSum[n, 1 &, !PrimeQ[#-1] &]; Array[a, 100] (* Amiram Eldar, Apr 13 2024 *)

PARI

a(n) = sumdiv(n, d, !isprime(d-1)); \\ Amiram Eldar, Apr 13 2024

Formula

a(n) = A000005(n) - A072627(n) < A000005(n).