login
Multiplicative projection of odd part of n.
1

%I #13 Nov 12 2022 05:25:48

%S 1,1,3,1,5,3,7,1,6,5,11,3,13,7,15,1,17,6,19,5,21,11,23,3,10,13,9,7,29,

%T 15,31,1,33,17,35,6,37,19,39,5,41,21,43,11,30,23,47,3,14,10,51,13,53,

%U 9,55,7,57,29,59,15,61,31,42,1,65,33,67,17,69,35,71,6,73,37,30

%N Multiplicative projection of odd part of n.

%H Amiram Eldar, <a href="/A341042/b341042.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.

%F a(n) = A000026(A000265(n)).

%F a(n) = A000026(n) if n odd, a(n) = a(n/2) if n even.

%F From _Amiram Eldar_, Nov 12 2022: (Start)

%F Multiplicative with a(2^e) = 1 and a(p^e) = e*p for p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (6*zeta(2)^2/17) * Product_{p prime} (1 - 3/p^2 + 2/p^3 + 1/p^4 - 1/p^5) = 0.2947570019... . (End)

%e a(54) = a(2 * 3^3) = 3 * 3 = 9.

%p a:= n-> mul(`if`(i[1]=2, 1, i[1]*i[2]), i=ifactors(n)[2]):

%p seq(a(n), n=1..75); # _Alois P. Heinz_, Feb 03 2021

%t a[n_] := Times @@ (#[[1]] #[[2]] & /@ FactorInteger[n/2^IntegerExponent[n, 2]]); Table[a[n], {n, 75}]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 1, f[i,1] * f[i,2]));} \\ _Amiram Eldar_, Nov 12 2022

%Y Cf. A000026, A000079 (positions of 1's), A000265, A056911 (fixed points), A204455.

%K nonn,mult

%O 1,3

%A _Ilya Gutkovskiy_, Feb 03 2021