A341526 Numerator of ratio n*sigma(A003961(n)) / sigma(n)*A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.

1, 8, 9, 52, 20, 4, 21, 64, 279, 160, 77, 26, 117, 28, 6, 1936, 170, 248, 114, 1040, 189, 308, 115, 32, 1425, 104, 1053, 26, 464, 16, 589, 1664, 231, 1360, 10, 124, 777, 304, 1053, 1280, 902, 42, 516, 22, 372, 230, 423, 968, 343, 3800, 17, 676, 530, 468, 110, 224, 513, 3712, 1829, 104, 2074, 589, 5859, 69952, 780, 154

Offset: 1

Author

Antti Karttunen, Feb 16 2021

Keywords

• nonn
• frac

Comments

Like the ratios sigma(n)/n, A003973(n)/A003961(n) and A003961(n)/n, also the ratio r(n) = A341528(n)/A341529(n) is multiplicative: if gcd(x,y) = 1, r(x*y) = r(x)*r(y).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8191
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A003973, A017665, A017666, A336849, A341525, A341528, A341529, A341530.

Cf. A341527 (denominators).

Cf. A341626 (same sequence as applied onto prime shift array A246278).

Programs

Mathematica

f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator[n*DivisorSigma[1, (gn = g[n])]/(DivisorSigma[1, n] * gn)]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)

PARI

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341526(n) = { my(s=A003961(n)); numerator((sigma(s)*n)/(sigma(n)*s)); };

Formula

a(n) = A341528(n) / A341530(n) = A341528(n) / gcd(A341528(n), A341529(n)).

For all n > 1, a(n) < A341527(n).