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A359593 Multiplicative with a(p^e) = 1 if p divides e, p^e otherwise.

%I A359593 #22 Nov 22 2023 10:47:29
%S A359593 1,2,3,1,5,6,7,8,9,10,11,3,13,14,15,1,17,18,19,5,21,22,23,24,25,26,1,
%T A359593 7,29,30,31,32,33,34,35,9,37,38,39,40,41,42,43,11,45,46,47,3,49,50,51,
%U A359593 13,53,2,55,56,57,58,59,15,61,62,63,1,65,66,67,17,69,70,71,72,73,74,75,19,77,78,79,5,81,82,83,21
%N A359593 Multiplicative with a(p^e) = 1 if p divides e, p^e otherwise.
%C A359593 Each term a(n) is a multiple of both A083346(n) and A327938(n).
%H A359593 Antti Karttunen, <a href="/A359593/b359593.txt">Table of n, a(n) for n = 1..16384</a>
%F A359593 a(n) = n / A359594(n).
%F A359593 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p^(p-1)*(p-1)/(p^(2*p)-1)) = 0.4225104173... . - _Amiram Eldar_, Jan 11 2023
%t A359593 f[p_, e_] := If[Divisible[e, p], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 09 2023 *)
%o A359593 (PARI) A359593(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^(f[k,2]*!!(f[k, 2]%f[k, 1]))); };
%o A359593 (Python)
%o A359593 from math import prod
%o A359593 from sympy import factorint
%o A359593 def A359593(n): return prod(p**e for p, e in factorint(n).items() if e%p) # _Chai Wah Wu_, Jan 10 2023
%Y A359593 Cf. A072873 (positions of 1's), A359594.
%Y A359593 Cf. also A083346, A327938, A342014, A359552.
%K A359593 nonn,mult
%O A359593 1,2
%A A359593 _Antti Karttunen_, Jan 09 2023