A369890 The number of divisors of the largest divisor of n whose exponents in its prime factorization are all powers of 2.

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 5, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset: 1

Author

Amiram Eldar, Feb 15 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A369015 at n = 32.

Links

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

Crossrefs

Cf. A000005, A000079, A053644, A138302, A353897, A369015.

Programs

Mathematica

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

PARI

a(n) = vecprod(apply(x -> 2^logint(x, 2) + 1, factor(n)[, 2]));

Formula

a(n) = A000005(A353897(n)).

Multiplicative with a(p^e) = A053644(e) + 1.

a(n) = 2 if and only if n is prime.

a(n) <= A000005(n), with equality if and only if n is in A138302.