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

1,2

Links

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

Formula

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000005, A072627.

Sequence in context: A029417 A029237 A078641 * A343901 A193386 A068212

Adjacent sequences: A072625 A072626 A072627 * A072629 A072630 A072631

Keyword

nonn

Author

Labos Elemer, Jun 28 2002

Status

approved