A372381 The number of divisors of the largest divisor of n whose number of divisors is a power of 2.

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

Offset

1,2

Comments

First differs from A286324 at n = 32, and from A331109 at n = 64.

Also, the number of infinitary divisors of the largest divisor of n whose number of divisors is a power of 2.

Links

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

Formula

Multiplicative with a(p^e) = 2^floor(log_2(e+1)).

a(n) = A000005(A372379(n)).

a(n) = A037445(A372379(n)).

a(n) = A000005(n) if and only if n is in A036537.

a(n) <= A372380(n), with equality if and only if n is cubefree (A004709).

Mathematica

f[p_, e_] := 2^Floor[Log2[e + 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> 2^exponent(x+1), factor(n)[, 2]));
(Python)
from math import prod
from sympy import factorint
def A372381(n): return prod(1<<(e+1).bit_length()-1 for e in factorint(n).values()) # Chai Wah Wu, Apr 30 2024

Crossrefs

Cf. A000005, A004709, A036537, A037445, A372379, A372380.

Cf. A286324, A331109.

Sequence in context: A335385 A037445 A318307 * A331109 A286324 A318472

Adjacent sequences: A372378 A372379 A372380 * A372382 A372383 A372384

Keyword

nonn,easy,mult

Author

Amiram Eldar, Apr 29 2024

Status

approved