A118123 a(n) = number of k's such that prime(n+1) = prime(n) + (prime(n) mod k).

0, 0, 1, 0, 2, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 1, 3, 2, 4, 3, 1, 4, 3, 3, 2, 5, 4, 7, 6, 2, 2, 2, 7, 2, 5, 2, 1, 2, 3, 1, 3, 3, 7, 6, 7, 2, 1, 2, 8, 7, 1, 3, 5, 4, 1, 1, 3, 2, 6, 5, 5, 3, 2, 3, 2, 2, 4, 2, 7, 6, 1, 6, 2, 1, 6, 3, 2, 2, 2, 5, 3, 2, 7, 3, 6, 3, 6, 2, 7, 6, 5, 2, 6, 5, 10, 3, 2, 3, 2, 2, 2, 3, 1, 9, 2

Offset: 1

Author

Rémi Eismann and Robert G. Wilson v, May 12 2006

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A117078, A117563.

Programs

Mathematica

f[n_] := If[n == 1, 0, Block[{p = Prime@n, np = Prime[n + 1]}, Length@Select[Divisors[2p - np], # >= np - p &]]]; Array[f, 105]
nk[n_]:=Count[Mod[n,Range[n-1]],?(#==NextPrime[n]-n&)]; nk/@Prime[ Range[ 110]] (* _Harvey P. Dale, May 27 2016 *)

PARI

A118123(n)={my(d=prime(n+1)-n=prime(n)); sumdiv(n-d,k,k>d)}

Formula

a(n) = # { k>0 | prime(n+1) - prime(n) = prime(n) % k }, where p % k is the remainder of p divided by k.

Extensions

Edited by M. F. Hasler, Nov 07 2009