A355933 -id:A355933 - cp's OEIS frontend

A355934 a(n) = sigma(n) / gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

1, 3, 2, 7, 3, 1, 2, 3, 13, 9, 6, 14, 7, 1, 1, 31, 9, 39, 5, 21, 4, 9, 4, 1, 31, 7, 10, 14, 15, 3, 16, 9, 4, 27, 1, 7, 19, 5, 14, 9, 21, 1, 11, 6, 39, 3, 8, 62, 3, 31, 3, 49, 9, 5, 9, 1, 5, 45, 30, 7, 31, 12, 26, 127, 7, 3, 17, 63, 8, 3, 36, 39, 37, 19, 62, 35, 4, 7, 20, 93, 11, 63, 14, 28, 27, 11, 5, 9, 45, 117

Offset: 1

Antti Karttunen, Jul 22 2022

Denominator of ratio A003973(n) / A000203(n). See comments in A355933.

Cf. A000203, A003961, A003973, A355932, A355933 (numerators), A355940, A355941 (positions of 1's).

Cf. also A336849, A349162.

f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n] / DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Jul 22 2022 *)

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355934(n) = { my(u=sigma(n)); (u/gcd(A003973(n), u)); };

a(n) = A000203(n) / A355932(n) = A000203(n) / gcd(A000203(n), A003973(n)).

A355932 a(n) = gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 4, 5, 1, 2, 2, 2, 2, 24, 24, 1, 2, 1, 4, 2, 8, 4, 6, 60, 1, 6, 4, 4, 2, 24, 2, 7, 12, 2, 48, 13, 2, 12, 4, 10, 2, 96, 4, 14, 2, 24, 6, 2, 19, 3, 24, 2, 6, 24, 8, 120, 16, 2, 2, 24, 2, 8, 4, 1, 12, 48, 4, 2, 12, 48, 2, 5, 2, 6, 2, 4, 24, 24, 4, 2, 11, 2, 6, 8, 4, 12, 24, 20, 2, 2, 8, 6, 4, 72, 24

Offset: 1

Antti Karttunen, Jul 22 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A003973, A355933, A355934.

Cf. also A336850, A342671.

f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := GCD[DivisorSigma[1, n], Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jul 22 2022 *)

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355932(n) = gcd(sigma(n), A003973(n));

a(n) = gcd(A000203(n), A003973(n)).
a(n) = A003973(n) / A355933(n).
a(n) = A000203(n) / A355934(n).

cp's OEIS Frontend

A355934 a(n) = sigma(n) / gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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