A368542 The number of divisors of n whose prime factors are all Mersenne primes (A000668).

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

Offset: 1

Author

Amiram Eldar, Dec 29 2023

Keywords

• nonn
• easy
• mult

Comments

The number of terms of A056652 U {1} that divide n.

Links

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

Crossrefs

Cf. A000005, A000668, A056652, A161790.

Programs

Mathematica

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], # + 1 == 2^IntegerExponent[# + 1, 2] &]; f[p_, e_] := If[q[p], e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f=factor(n)); prod(i=1, #f~, if((f[i,1]+1) >> valuation(f[i,1]+1, 2) == 1 , f[i,2] + 1, 1))};

Formula

Multiplicative with a(p^e) = e+1 if p is a Mersenne prime (A000668), and 1 otherwise.

a(n) >= 1, with equality if and only if n is in A161790.

a(n) <= A000005(n), with equality if and only if n is in A056652 U {1}.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/Product_{k>=1} (1 - 1/A000668(k)) = 1.82292512097260346512... .