A372470 -id:A372470 - cp's OEIS frontend

A372468 The number of divisors of the numbers whose number of divisors is a power of 2 (A036537).

1, 2, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 2, 4, 2, 8, 4, 8, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 8, 2, 4, 4, 4, 4, 2, 2, 8, 2, 8, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 16, 4, 4, 4

Offset: 1

Amiram Eldar, May 02 2024

Also, the number of infinitary divisors of the numbers whose number of divisors is a power of 2.

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

Cf. A000005, A036537, A037445, A372469, A372470.

Select[DivisorSigma[0, Range[300]], # == 2^IntegerExponent[#, 2] &]

lista(n) = select(x -> x >> valuation(x, 2) == 1, vector(n, i, numdiv(i)));

a(n) = A000005(A036537(n)).
a(n) = A037445(A036537(n)).
a(n) = A372470(A036537(n)).
a(n) = 2^A372469(n).

A372504 Multiplicative with a(p^e) = e if e is a power of 2, and 0 otherwise.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 0, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 4, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, May 04 2024

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

Cf. A005361, A048298, A138302, A209229, A355823, A368473, A372470.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x == 1 << valuation(x, 2), x, 0), factor(n)[, 2]));

Multiplicative with a(p^e) = A048298(e) = A209229(e) * e.
a(n) = A355823(n) * A005361(n).
a(A138302(n)) = A005361(A138302(n)) = A368473(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.31285540951965780409..., where f(x) = (1-x) * (1 + Sum_{k>=0} 2^k * x^(2^k)).

cp's OEIS Frontend

A372468 The number of divisors of the numbers whose number of divisors is a power of 2 (A036537).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula