A372328 a(n) is the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1

Offset

1,4

Comments

First differs from A331738 at n = 32.

The largest divisor d of n that is infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.

Links

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

Formula

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

a(n) = A372329(n)/n.

a(n) = 1 if and only if n is in A036537.

a(n) <= n, with equality if and only if n = 1.

Mathematica

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

Prog

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

Crossrefs

Cf. A036537, A064379, A077609, A331738, A372329.

Similar sequences: A365297, A365298, A365637, A365685, A367931, A367932.

Sequence in context: A071974 A056622 A331738 * A306333 A237983 A335977

Adjacent sequences: A372325 A372326 A372327 * A372329 A372330 A372331

Keyword

nonn,easy,mult

Author

Amiram Eldar, Apr 28 2024

Status

approved