A072627 - OEIS

A072627

Number of divisors d of n such that d-1 is prime.

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

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OFFSET

1,6

COMMENTS

Inverse Möbius transform of A010051 (when it is shifted one step right). - R. J. Mathar, Jan 25 2009, comment in parenthesis added by Antti Karttunen, Dec 27 2018

If n == 3 (mod 6) then a(n)=1; a(n) = 0 for all other odd n. - Robert Israel, Dec 27 2018

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

If p is prime <> 3, then divisors(p)-1 = {1,p}-1 = {0,p-1}, so a(p) = 0.

EXAMPLE

n=240: a(240)=12 because primes of -1+d form are: {2,3,5,7,11,19,23,29,47,59,79,239}. These and only these divisors are present in any InvSigma of n, like:InvSig[240]= {114,135,158,177,203,209,239} with {2,3,19,3,5,2,79,3,59,7,29,11,19,239} p-divisors.

MAPLE

f:= n -> nops(select(t -> isprime(t-1), numtheory:-divisors(n))):

map(f, [$1..100]); # Robert Israel, Dec 27 2018

MATHEMATICA

di[x_] := Divisors[x] dp[x_] := Part[di[x], Flatten[Position[PrimeQ[ -1+di[x]], True]]]-1 Table[Length[dp[w]], {w, 1, 128}]

Table[Count[Divisors[n], _?(PrimeQ[#-1]&)], {n, 110}] (* Harvey P. Dale, Jul 04 2021 *)

PROG

(Haskell)

a072627 = length . filter ((== 1) . a010051 . (subtract 1)) . a027749_row

-- Reinhard Zumkeller, Oct 01 2012

(PARI) a(n) = sumdiv(n, d, isprime(d-1)); \\ Michel Marcus, Dec 27 2018

CROSSREFS