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A378521 Möbius transform of A048673.

%I A378521 #8 Nov 30 2024 23:47:01
%S A378521 1,1,2,3,3,4,5,9,10,6,6,12,8,10,12,27,9,20,11,18,20,12,14,36,21,16,50,
%T A378521 30,15,24,18,81,24,18,30,60,20,22,32,54,21,40,23,36,60,28,26,108,55,
%U A378521 42,36,48,29,100,36,90,44,30,30,72,33,36,100,243,48,48,35,54,56,60,36,180,39,40,84,66,60,64,41,162,250
%N A378521 Möbius transform of A048673.
%H A378521 Antti Karttunen, <a href="/A378521/b378521.txt">Table of n, a(n) for n = 1..20000</a>
%H A378521 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%F A378521 a(n) = Sum_{d|n} A008683(n/d)*A048673(d).
%F A378521 a(n) = A072451(A048673(n)).
%F A378521 a(n) = A055034(A003961(n)) = A349136(A003961(n)).
%F A378521 For n > 1, a(n) = A003972(n)/2 = A000010(A003961(n))/2.
%o A378521 (PARI)
%o A378521 A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
%o A378521 A378521(n) = sumdiv(n,d,moebius(n/d)*A048673(d));
%o A378521 (PARI)
%o A378521 A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A378521 A378521(n) = if(1==n,n,eulerphi(A003961(n))/2);
%Y A378521 Cf. A000010, A003961, A003972, A008683, A048673, A055034, A072451, A349136.
%K A378521 nonn
%O A378521 1,3
%A A378521 _Antti Karttunen_, Nov 30 2024