A355933 a(n) = A003973(n) / gcd(sigma(n), A003973(n)), where A003973(n) = sigma(A003961(n)) and A003961 is fully multiplicative with a(p) = nextprime(p).

1, 4, 3, 13, 4, 2, 3, 8, 31, 16, 7, 39, 9, 2, 2, 121, 10, 124, 6, 52, 9, 14, 5, 4, 57, 12, 39, 39, 16, 8, 19, 52, 7, 40, 2, 31, 21, 8, 27, 32, 22, 3, 12, 13, 124, 5, 9, 363, 7, 76, 5, 117, 10, 26, 14, 4, 9, 64, 31, 26, 34, 19, 93, 1093, 12, 7, 18, 130, 15, 8, 37, 248, 40, 28, 171, 78, 7, 18, 21, 484, 71, 88, 15, 117

Offset: 1

Author

Antti Karttunen, Jul 22 2022

Keywords

• nonn
• frac

Comments

Numerator of ratio A003973(n) / A000203(n). This sequence gives the numerators when presented in its lowest terms, while A355934 gives the denominators. As both A000203 and A003973 are multiplicative sequences, their ratio is also: 1, 4/3, 3/2, 13/7, 4/3, 2/1, 3/2, 8/3, 31/13, 16/9, 7/6, 39/14, 9/7, 2/1, 2/1, 121/31, 10/9, 124/39, 6/5, etc.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A003973, A355932, A355934 (denominators).

Cf. also A341525, A349161.

Programs

Mathematica

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

PARI

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355933(n) = { my(u=A003973(n)); (u/gcd(sigma(n), u)); };

Formula

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