A372330 The number of divisors of the smallest multiple of n whose number of divisors is a power of 2 (A036537).

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

Offset: 1

Author

Amiram Eldar, Apr 28 2024

Keywords

• nonn
• easy
• mult

Comments

Also, the number of infinitary divisors of the smallest multiple of n whose number of divisors is a power of 2.

Links

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

Crossrefs

Cf. A000005, A036537, A037445, A372329.

Programs

Mathematica

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

PARI

s(n) = {my(e=logint(n + 1, 2)); if(n + 1 == 2^e, n + 1, 2^(e+1))};
a(n) = {my(f=factor(n)); prod(i=1, #f~, s(f[i, 2]))};

Formula

a(n) = A000005(A372329(n)).

a(n) = A037445(A372329(n)).

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