A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 128, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 128, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 384, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Author

Amiram Eldar, Apr 28 2024

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A036537, A372328.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193, A356194.

Differs from A102631 at n = 8, 24, 27, 32, 40, 54, 56, 64, ... .

Programs

Mathematica

f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - 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, 2^(e+1) - 1)};
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+1)) - 1).

a(n) = n * A372328(n).

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

a(n) <= n^2, with equality if and only if n = 1.