A295657 Multiplicative with a(p^e) = p^floor((e-1)/2).

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

Offset: 1

Author

Antti Karttunen, Nov 28 2017

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A000188, A003557, A004526, A020639, A028234, A067029, A295658, A295659.

Cf. A004709 (positions of ones), A082020, A212793.

Cf. also A008834, A053150, A061704.

Programs

Mathematica

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> p^Floor[(e - 1)/2]] &, 105] (* Michael De Vlieger, Nov 28 2017 *)

PARI

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

Formula

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

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

a(n) = 1 iff A212793(n) = 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15/Pi^2 = 1.519817... (A082020). - Amiram Eldar, Nov 30 2022