A365637 -id:A365637 - cp's OEIS frontend

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

Amiram Eldar, Apr 28 2024

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.

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

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

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

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

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]))};

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.

A365636 a(n) is the smallest multiple of n that is a term of A072873.

Original entry on oeis.org

1, 4, 27, 4, 3125, 108, 823543, 16, 27, 12500, 285311670611, 108, 302875106592253, 3294172, 84375, 16, 827240261886336764177, 108, 1978419655660313589123979, 12500, 22235661, 1141246682444, 20880467999847912034355032910567, 432, 3125, 1211500426369012, 27, 3294172

Offset: 1

Amiram Eldar, Sep 14 2023

Amiram Eldar, Table of n, a(n) for n = 1..388

Cf. A072873, A365637.

f[p_, e_] := p^(Ceiling[e/p]*p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,1] * ceil(f[i,2] / f[i,1])));}

Multiplicative with a(p^e) = p^(p*ceiling(e/p)).
a(n) = n * A365637(n).
a(n) >= n with equality if and only if n is in A072873.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^p-1)) = 1.86196549645040699446... .

Data, formulas and codes corrected by Amiram Eldar, Feb 15 2024

cp's OEIS Frontend

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).

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