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%I #10 May 13 2022 09:53:19
%S 1,1,1,4,1,1,1,32,9,1,1,48,1,1,15,512,1,9,1,80,21,1,1,9216,25,1,9,112,
%T 1,15,1,16384,33,1,35,15552,1,1,39,25600,1,21,1,176,135,1,1,7077888,
%U 49,25,51,208,1,9,55,50176,57,1,1,864000,1,1,189,1048576,65,33,1,272,69
%N Product of the divisors of n whose arithmetic derivative is even.
%F a(n) = Product_{d|n} d^((1+d') mod 2), where d' is the arithmetic derivative of d.
%e a(8) = 32; the product of the divisors of 8 whose arithmetic derivative is even is 4*8 = 32.
%t d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := Product[k^Mod[d[k] + 1, 2], {k, Divisors[n]}]; Array[a, 60] (* _Amiram Eldar_, May 13 2022 *)
%o (PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o a(n) = vecprod(select(x->!(ad(x) % 2), divisors(n))); \\ _Michel Marcus_, May 13 2022
%Y Cf. A003415 (n'), A353975.
%K nonn
%O 1,4
%A _Wesley Ivan Hurt_, May 12 2022
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