A365297 a(n) is the smallest number k such that k*n is a number whose prime factorization exponents are all powers of 2 (A138302).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Aug 31 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A270419 at n = 128.

Links

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

Crossrefs

Cf. A138302, A270419, A356194.

Programs

Mathematica

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

PARI

s(n) = {my(e = logint(n, 2)); if(n == 2^e, 0, 2^(e+1) - n)};
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]))};

Formula

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

a(n) = A356194(n)/n.

a(n) = 1 if and only if n is in A138302.