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%I #23 May 06 2024 21:01:13
%S 1,1,1,1,1,3,3,1,1,1,1,1,3,1,1,1,1,3,3,1,1,1,1,3,1,1,1,43,1,3,5,1,1,1,
%T 1,1,3,1,1,1,1,3,3,1,1,5,1,1,1,1,1,1,1,3,19,1,1,1,1,3,5,1,1,1,1,3,3,5,
%U 7,1,1,3,1,1,1,1,1,3,3,1,1,1,1,1,1,1,1,1,1,3,5,1,1,1,1,3,3,1,1,1,1,3,3,1,1
%N a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).
%C Terms a(n) larger than 1 and equal to A252748(n) occur at n = 6, 28, 69, 91, 496, ..., see A326134. See also A349753.
%C Records 1, 3, 43, 45, 2005, 79243, ... occur at n = 1, 6, 28, 360, 496, 8128, ...
%H Antti Karttunen, <a href="/A326057/b326057.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = gcd(A252748(n), A286385(n)) = gcd(A003961(n) - 2n, A003961(n) - A000203(n)).
%F a(n) = gcd(A252748(n), A033879(n)) = gcd(A286385(n), A033879(n)). [Also A033880 can be used] - _Antti Karttunen_, May 06 2024
%t Array[GCD[#3 - #1, #3 - #2] & @@ {2 #, DivisorSigma[1, #], Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 78] (* _Michael De Vlieger_, Feb 22 2021 *)
%o (PARI)
%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A252748(n) = (A003961(n) - (2*n));
%o A286385(n) = (A003961(n) - sigma(n));
%o A326057(n) = gcd(A252748(n), A286385(n));
%Y Cf. A000203, A000360, A003961, A033879, A033880, A252748, A286385, A326134.
%Y Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326060, A326062, A349753.
%K nonn
%O 1,6
%A _Antti Karttunen_, Jun 06 2019
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