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A072627 Number of divisors d of n such that d-1 is prime.

%I A072627 #36 Jul 04 2021 19:02:11
%S A072627 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,
%T A072627 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,
%U A072627 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
%N A072627 Number of divisors d of n such that d-1 is prime.
%C A072627 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
%C A072627 If n == 3 (mod 6) then a(n)=1; a(n) = 0 for all other odd n. - _Robert Israel_, Dec 27 2018
%H A072627 Reinhard Zumkeller, <a href="/A072627/b072627.txt">Table of n, a(n) for n = 1..10000</a>
%F A072627 If p is prime <> 3, then divisors(p)-1 = {1,p}-1 = {0,p-1}, so a(p) = 0.
%e A072627 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.
%p A072627 f:= n -> nops(select(t -> isprime(t-1), numtheory:-divisors(n))):
%p A072627 map(f, [$1..100]); # _Robert Israel_, Dec 27 2018
%t A072627 di[x_] := Divisors[x] dp[x_] := Part[di[x], Flatten[Position[PrimeQ[ -1+di[x]], True]]]-1 Table[Length[dp[w]], {w, 1, 128}]
%t A072627 Table[Count[Divisors[n],_?(PrimeQ[#-1]&)],{n,110}] (* _Harvey P. Dale_, Jul 04 2021 *)
%o A072627 (Haskell)
%o A072627 a072627 = length . filter ((== 1) . a010051 . (subtract 1)) . a027749_row
%o A072627 -- _Reinhard Zumkeller_, Oct 01 2012
%o A072627 (PARI) a(n) = sumdiv(n, d, isprime(d-1)); \\ _Michel Marcus_, Dec 27 2018
%Y A072627 Cf. A000203, A000005.
%Y A072627 Cf. also A067513.
%K A072627 nonn
%O A072627 1,6
%A A072627 _Labos Elemer_, Jun 28 2002