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

1,8

Links

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

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

Mathematica

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

Prog

(Scheme, with memoization-macro definec) (definec (A295657 n) (if (= 1 n) 1 (* (expt (A020639 n) (A004526 (- (A067029 n) 1))) (A295657 (A028234 n)))))
(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

Crossrefs

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

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

Cf. also A008834, A053150, A061704.

Sequence in context: A325988 A328856 A053150 * A163379 A006466 A316439

Adjacent sequences: A295654 A295655 A295656 * A295658 A295659 A295660

Keyword

nonn,mult

Author

Antti Karttunen, Nov 28 2017

Status

approved