A341530 -id:A341530 - cp's OEIS frontend

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

Antti Karttunen, Feb 16 2021

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

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

Cf. A336849 (denominators).

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 *)

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)); };

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

A342670 a(n) = gcd(nsigma(A064989(n)), sigma(n)A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 2, 1, 1, 2, 4, 4, 2, 12, 36, 1, 2, 6, 2, 2, 2, 4, 4, 24, 1, 6, 5, 56, 6, 72, 2, 1, 24, 2, 120, 28, 2, 12, 4, 10, 2, 12, 2, 4, 36, 8, 4, 8, 1, 1, 18, 2, 6, 30, 8, 24, 2, 6, 6, 144, 2, 12, 2, 1, 12, 144, 2, 14, 12, 240, 4, 12, 2, 2, 9, 4, 336, 24, 2, 2, 1, 2, 4, 56, 4, 12, 24, 4, 6, 72, 56, 8, 2, 8, 360

Offset: 1

Antti Karttunen, Mar 24 2021

nonn

Cf. A000203, A064989, A326041, A341530 [ = a(A003961(n))], A342661, A342662, A342663, A342664.

A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));
A342662(n) = (sigma(n)*A064989(n));
A342670(n) = gcd(A342661(n), A342662(n));

a(n) = gcd(A342661(n), A342662(n)).

a(n) = gcd(n*A000203(A064989(n)), A000203(n)*A064989(n)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A342670 a(n) = gcd(nsigma(A064989(n)), sigma(n)A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A342670 a(n) = gcd(n*sigma(A064989(n)), sigma(n)*A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A342670 a(n) = gcd(nsigma(A064989(n)), sigma(n)A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.