A295658 Multiplicative with a(p^e) = p^max(0,(floor(e/2)-1)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Nov 28 2017

Keywords

• nonn
• mult

Comments

a(n) differs from A053164(n) = A000188(A000188(n)) for the first time at n=64, where a(64) = 4, while A053164(64) = 2.

Examples

			For n = 64 = 2^6, a(64) = 2^(floor(6/2)-1) = 2^2 = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000188, A003557, A004526, A020639, A028234, A053164, A067029, A295657.

Cf. A046100 (positions of ones), A157289.

Programs

Mathematica

f[p_, e_] := p^Max[0, Floor[e/2-1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 30 2022 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^max(0, floor(f[i,2]/2-1)));} \\ Amiram Eldar, Nov 30 2022

Formula

a(1) = 1; for n > 1, a(n) = A020639(n)^max(0,A004526(A067029(n))-1) * a(A028234(n)).

a(n) = A003557(A000188(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(6) = 1.181564... (A157289). - Amiram Eldar, Nov 30 2022