A368333 The largest term of A054744 that divide n.

1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 1, 64, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1

Offset

1,4

Comments

The largest divisor d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

Links

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

Formula

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = p^e if e >= p.

A034444(a(n)) = A368334(n).

a(n) >= 1, with equality if and only if n is in A048103.

a(n) <= n, with equality if and only if n is in A054744.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^(p*s) + 1/p^(p*(s-1)) + 1/p^((p+1)*s-1) - 1/p^((p+1)*(s-1)+1)).

Mathematica

f[p_, e_] := If[e < p, 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < f[i, 1], 1, f[i, 1]^f[i, 2])); }

Crossrefs

Cf. A034444, A048103, A054744, A327939, A368329, A368332, A368334, A368335.

Sequence in context: A073937 A074058 A308552 * A088440 A300253 A212173

Adjacent sequences: A368330 A368331 A368332 * A368334 A368335 A368336

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 21 2023

Status

approved