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

1,2

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)

Formula

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

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

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 *)

Prog

(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)); };

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).

Sequence in context: A103315 A175931 A270013 * A042847 A121330 A152189

Adjacent sequences: A341523 A341524 A341525 * A341527 A341528 A341529

Keyword

nonn,frac

Author

Antti Karttunen, Feb 16 2021

Status

approved