Amiram Eldar, <a href="/A368329/b368329_1.txt">Table of n, a(n) for n = 1..10000</a>

proposed

approved

editing

proposed

Amiram Eldar, <a href="/A368329/b368329_1.txt">Table of n, a(n) for n = 1..10000</a>

allocated for Amiram EldarThe largest term of A054743 that divide n.

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 81, 1, 1, 1, 1

1,8

First differs from A360540 at n = 27.

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).

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

A034444(a(n)) = A368330(n).

a(n) >= 1, with equality if and only if n is in A207481.

a(n) <= n, with equality if and only if n is in A054743.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^((p+2)*s-1) - 1/p^((p+2)*(s-1)+1) - 1/p^((p+1)*s) + 1/p^((p+1)*(s-1))).

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

(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])); }

Cf. A054743, A034444, A207481, A327939, A360540, A368329, A368330, A368331, A368333.

allocated

nonn,easy,mult

Amiram Eldar, Dec 21 2023

approved

editing

allocated for Amiram Eldar

allocated

approved