A350072 a(n) = sigma(n^2) / 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.

1, 7, 13, 31, 31, 91, 57, 127, 121, 31, 133, 403, 183, 133, 403, 511, 307, 847, 381, 961, 741, 931, 553, 1651, 781, 427, 1093, 589, 871, 403, 993, 2047, 133, 2149, 1767, 3751, 1407, 889, 2379, 3937, 1723, 1729, 1893, 4123, 3751, 3871, 2257, 6643, 2801, 781, 3991, 1891, 2863, 7651, 589, 2413, 4953, 6097, 3541, 12493

Offset: 1

Author

Antti Karttunen, Dec 12 2021

Keywords

• nonn

Comments

Conjecture: There are no 1's after the initial term. Remark: If there were some k = x^2 > 1, for which a(x) = 1, then sigma(k) would be a divisor of A003961(k). In other words, d = A350073(k) = A064989(sigma(k)) would be a divisor of k. Then, if that divisor were also a unitary divisor [with gcd(d,k/d) = 1], it would need to satisfy the equation sigma(k) = sigma(d) * sigma(k/d) = sigma(A064989(sigma(k))) * sigma(k/A064989(sigma(k))), because sigma is a multiplicative function. (Minor correction by Antti Karttunen, Jul 11 2023)

Note that if d = A064989(sigma(k)) were a unitary divisor of a square k, then sigma(k) would also be a square, the cases which are quite rare (see A008848 and A336547). Also compare to A349756. - Antti Karttunen, Jul 24 2022

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A008848, A064989, A065764, A349162, A350071, A350073.

See also A069070, A336547, A349756, A364131.

Programs

Mathematica

f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^(2*e); a[1] = 1; a[n_] := (s = Times @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, Times @@ f2 @@@ f]; Array[a, 60] (* Amiram Eldar, Dec 12 2021 *)

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))); };
A350072(n) = A349162(n^2);

Formula

a(n) = A349162(n^2).

a(n) = A065764(n) / A350071(n).