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a(n) = gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.
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%I #12 Dec 15 2021 21:43:40

%S 1,1,1,1,1,1,1,1,1,7,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1,7,1,1,13,1,

%T 1,1,1,3,1,1,1,3,1,1,1,1,1,1,1,7,1,3,1,1,7,3,1,1,1,1,1,3,121,1,1,13,1,

%U 1,1,21,1,1,1,3,1,3,1,3,1,7,1,1,1,3,1,3,1,1,1,7,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1

%N a(n) = gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

%H Antti Karttunen, <a href="/A350071/b350071.txt">Table of n, a(n) for n = 1..20000</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) = A342671(n^2).

%F a(n) = A065764(n) / A350072(n).

%t f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^(2*e); a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; Array[a, 100] (* _Amiram Eldar_, Dec 12 2021 *)

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A342671(n) = gcd(sigma(n), A003961(n));

%o A350071(n) = A342671(n^2);

%Y Cf. A000203, A000290, A003961, A065764, A350072.

%K nonn

%O 1,10

%A _Antti Karttunen_, Dec 12 2021