A348993 a(n) = A064989(sigma(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

1, 1, 1, 5, 2, 1, 1, 3, 11, 2, 2, 5, 5, 1, 2, 29, 4, 11, 3, 1, 1, 2, 2, 1, 29, 5, 1, 5, 6, 2, 1, 5, 2, 4, 2, 55, 17, 3, 5, 3, 10, 1, 7, 5, 22, 2, 2, 29, 34, 29, 4, 25, 8, 1, 4, 3, 1, 6, 6, 1, 29, 1, 11, 113, 2, 2, 13, 5, 2, 2, 4, 11, 31, 17, 29, 15, 2, 5, 3, 29, 49, 10, 10, 5, 8, 7, 2, 3, 12, 22, 5, 5, 1, 2, 6, 5

Offset: 1

Author

Antti Karttunen, Nov 10 2021

Keywords

• nonn
• look

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)

Crossrefs

Cf. A000203, A000265, A003961, A064989, A161942, A342671, A348992, A349162, A349169 (gives odd k for which a(k) = A319627(k)).

Programs

Mathematica

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#1/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 96] (* Michael De Vlieger, Nov 11 2021 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };
A348993(n) = A064989(A349162(n));

Formula

a(n) = A064989(A349162(n)) = A064989(A348992(n)).