A341525 Numerator of A003973(n) / A003961(n).

1, 4, 6, 13, 8, 8, 12, 40, 31, 32, 14, 26, 18, 16, 48, 121, 20, 124, 24, 104, 72, 56, 30, 16, 57, 24, 156, 52, 32, 64, 38, 364, 84, 80, 96, 403, 42, 32, 108, 320, 44, 96, 48, 14, 248, 40, 54, 242, 133, 76, 24, 26, 60, 208, 16, 160, 144, 128, 62, 208, 68, 152, 372, 1093, 144, 112, 72, 260, 36, 128, 74, 248, 80, 56

Offset: 1

Author

Antti Karttunen, Feb 16 2021

Keywords

• nonn
• frac

Comments

Also numerator of the ratio (A341528(n)/A341529(n)) / (n/sigma(n)).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383
• Index entries for sequences computed from indices in prime factorization.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A000290 (positions of odd terms), A003961, A003973, A017665, A151800, A336850, A341526, A341527, A341528, A341529, A341530.

Cf. A336849 (denominators).

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/((p - 1)*p^e); g[p_, e_] := f[NextPrime[p], e]; a[1] = 1; a[n_] := Numerator[Times @@ g @@@ FactorInteger[n]]; 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
A341525(n) = { my(u=A003961(n), s=sigma(u)); (s/gcd(u, s)); };

Formula

a(n) = A017665(A003961(n)).

a(n) = A003973(n) / A336850(n) = A003973(n) / gcd(A003961(n), A003973(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} A341525(k)/A336849(k) = 1 / Product_{p prime} (1 - 1/(p*nextprime(p))) = 1.3766054560..., where nextprime(p) = A151800(p). - Amiram Eldar, Dec 28 2024