A326057 a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jun 06 2019

Keywords

• nonn

Comments

Terms a(n) larger than 1 and equal to A252748(n) occur at n = 6, 28, 69, 91, 496, ..., see A326134. See also A349753.

Records 1, 3, 43, 45, 2005, 79243, ... occur at n = 1, 6, 28, 360, 496, 8128, ...

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A000360, A003961, A033879, A033880, A252748, A286385, A326134.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326060, A326062, A349753.

Programs

Mathematica

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 *)

PARI

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A252748(n) = (A003961(n) - (2*n));
A286385(n) = (A003961(n) - sigma(n));
A326057(n) = gcd(A252748(n), A286385(n));

Formula

a(n) = gcd(A252748(n), A286385(n)) = gcd(A003961(n) - 2n, A003961(n) - A000203(n)).

a(n) = gcd(A252748(n), A033879(n)) = gcd(A286385(n), A033879(n)). [Also A033880 can be used] - Antti Karttunen, May 06 2024