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

1,6

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)

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

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 *)

Prog

(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));

Crossrefs

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

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

Sequence in context: A355604 A059790 A307156 * A130778 A353516 A351577

Adjacent sequences: A326054 A326055 A326056 * A326058 A326059 A326060

Keyword

nonn

Author

Antti Karttunen, Jun 06 2019

Status

approved