A349162 a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

1, 1, 4, 7, 6, 4, 8, 5, 13, 6, 12, 28, 14, 8, 24, 31, 18, 13, 20, 2, 32, 12, 24, 4, 31, 14, 8, 56, 30, 24, 32, 7, 48, 18, 48, 91, 38, 20, 56, 10, 42, 32, 44, 28, 78, 24, 48, 124, 57, 31, 72, 98, 54, 8, 72, 40, 16, 30, 60, 8, 62, 32, 104, 127, 12, 48, 68, 14, 96, 48, 72, 13, 74, 38, 124, 140, 96, 56, 80, 62, 121, 42

Offset

1,3

Comments

Denominator of ratio A003961(n) / A000203(n).

Small values are rare, but are not limited to the beginning. For example in range 1 .. 2^25, a(n) = 4 at n = 3, 6, 24, 792, 2720, 122944, 31307472.

Question: Would it be possible to prove that a(n) > 1 for all n > 2?

Obviously, 1's may occur only on squares & twice squares (A028982). See also comments in A350072. - Antti Karttunen, Feb 16 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..22968

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

Formula

a(n) = A000203(n) / A342671(n) = A000203(n) / gcd(A000203(n), A003961(n)).

Mathematica

Array[#1/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 82] (* Michael De Vlieger, Nov 11 2021 *)

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349162(n) = { my(s=sigma(n)); (s/gcd(s, A003961(n))); };

Crossrefs

Cf. A000203, A003961, A028982 (positions of odd terms), A319630, A336702, A342671, A348992 (the odd part), A348993, A349161 (numerators), A349163, A349164, A349627, A349628, A350072 [= a(n^2)].

Cf. also A349745, A351551, A351554.

Sequence in context: A112518 A228715 A351546 * A351547 A354828 A308366

Adjacent sequences: A349159 A349160 A349161 * A349163 A349164 A349165

Keyword

nonn,frac

Author

Antti Karttunen, Nov 09 2021

Status

approved