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A368542
The number of divisors of n whose prime factors are all Mersenne primes (A000668).
1
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
1,3
COMMENTS
The number of terms of A056652 U {1} that divide n.
LINKS
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... .
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]
PROG
(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))};
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 29 2023
STATUS
approved