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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.
7
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
1,2
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
FORMULA
a(n) = A349162(n^2).
a(n) = A065764(n) / A350071(n).
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 *)
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))); };
A350072(n) = A349162(n^2);
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 12 2021
STATUS
approved