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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).
4
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
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
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Apr 28 2024
STATUS
approved