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%I A348946 #10 Feb 11 2022 11:23:48
%S A348946 1,3,4,7,6,12,8,3,13,18,12,28,14,24,24,1,18,39,20,42,32,36,24,12,31,
%T A348946 42,2,56,30,72,32,3,48,54,48,1,38,60,56,18,42,96,44,84,78,72,48,4,57,
%U A348946 93,72,98,54,6,72,24,80,90,60,168,62,96,104,1,84,144,68,126,96,144,72,3,74,114,124,140,96,168,80,6,1,126
%N A348946 a(n) = gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.
%C A348946 This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 1 <> 91 = 7*13 = a(4)*a(9).
%H A348946 Antti Karttunen, <a href="/A348946/b348946.txt">Table of n, a(n) for n = 1..20000</a>
%H A348946 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348946 a(n) = gcd(A000203(n), A348944(n)).
%F A348946 a(n) = gcd(A000203(n), A348945(n)) = gcd(A348944(n), A348945(n));
%F A348946 a(n) = A348944(n) / A348947(n) = A000203(n) / A348948(n).
%t A348946 f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), (Times @@ f2 @@@ f + Times @@ f3 @@@ f)/2]; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)
%o A348946 (PARI)
%o A348946 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348946 A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
%o A348946 A348946(n) = gcd(sigma(n), ((1/2)*(A003959(n)+A034448(n))));
%Y A348946 Cf. A000203, A003959, A034448, A348944, A348945, A348947, A348948.
%Y A348946 Cf. also A344695, A348047, A348503, A348733.
%K A348946 nonn
%O A348946 1,2
%A A348946 _Antti Karttunen_, Nov 05 2021

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