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

%I A295658 #14 Nov 30 2022 08:34:52
%S A295658 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,
%T A295658 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,
%U A295658 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
%N A295658 Multiplicative with a(p^e) = p^max(0,(floor(e/2)-1)).
%C A295658 a(n) differs from A053164(n) = A000188(A000188(n)) for the first time at n=64, where a(64) = 4, while A053164(64) = 2.
%H A295658 Antti Karttunen, <a href="/A295658/b295658.txt">Table of n, a(n) for n = 1..20000</a>
%F A295658 a(1) = 1; for n > 1, a(n) = A020639(n)^max(0,A004526(A067029(n))-1) * a(A028234(n)).
%F A295658 a(n) = A003557(A000188(n)).
%F A295658 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
%e A295658 For n = 64 = 2^6, a(64) = 2^(floor(6/2)-1) = 2^2 = 4.
%t A295658 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 *)
%o A295658 (Scheme, with memoization-macro definec) (definec (A295658 n) (if (= 1 n) 1 (* (expt (A020639 n) (max 0 (+ -1 (A004526 (A067029 n))))) (A295658 (A028234 n)))))
%o A295658 (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
%Y A295658 Cf. A000188, A003557, A004526, A020639, A028234, A053164, A067029, A295657.
%Y A295658 Cf. A046100 (positions of ones), A157289.
%K A295658 nonn,mult
%O A295658 1,16
%A A295658 _Antti Karttunen_, Nov 28 2017