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A326297 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)^(k_j - 1)).

%I A326297 #51 Nov 01 2024 11:49:15
%S A326297 1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,4,1,4,1,1,1,1,1,1,1,
%T A326297 1,2,1,1,1,1,1,1,1,1,2,1,1,1,6,4,1,1,1,4,1,1,1,1,1,1,1,1,2,1,1,1,1,1,
%U A326297 1,1,1,2,1,1,4,1,1,1,1,1,8,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,6,2,4
%N A326297 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)^(k_j - 1)).
%H A326297 Antti Karttunen, <a href="/A326297/b326297.txt">Table of n, a(n) for n = 1..20000</a>
%H A326297 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A326297 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A326297 a(n) = A003958(n) / abs(A023900(n)) = abs(A325126(n)) / A007947(n).
%F A326297 Dirichlet g.f.: Product_{p prime} (1 + 1/(p^s - p + 1)). - _Amiram Eldar_, Dec 07 2023
%F A326297 a(n) = A003958(n)/A173557(n). - _Ridouane Oudra_, Oct 29 2024
%e A326297 a(98) = a(2 * 7^2) = (2 - 1)^(1 - 1) * (7 - 1)^(2 - 1) = 6.
%p A326297 seq(mul((p-1)^(padic[ordp](n,p)-1), p in numtheory[factorset](n)), n =1..100); # _Ridouane Oudra_, Oct 29 2024
%t A326297 a[n_] := If[n == 1, 1, Times @@ ((#[[1]] - 1)^(#[[2]] - 1) & /@ FactorInteger[n])]; Table[a[n], {n, 1, 100}]
%o A326297 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]--; f[k,2]--); factorback(f); \\ _Michel Marcus_, Mar 03 2020
%o A326297 (Python)
%o A326297 from math import prod
%o A326297 from sympy import factorint
%o A326297 def a(n): return prod((p-1)**(e-1) for p, e in factorint(n).items())
%o A326297 print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, Aug 30 2021
%Y A326297 Cf. A003557, A003958, A003959, A007947, A023900, A064478, A064549, A122132 (positions of 1's), A125131, A173557, A325126, A327564.
%K A326297 nonn,mult
%O A326297 1,9
%A A326297 _Ilya Gutkovskiy_, Mar 03 2020