A166120 -id:A166120 - cp's OEIS frontend

A166123 If n is prime, a(n) = 1; otherwise, a(n) is gcd(n, d) where d is the denominator of the (n-1)-th Bernoulli number.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Paul Curtz, Oct 07 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 1..21218

Cf. A166062, A027642.

Table[If[PrimeQ[n],1,GCD[n,Denominator[BernoulliB[n-1]]]],{n,100}] (* Harvey P. Dale, Sep 07 2017 *)

a(n)=if(isprime(n),1,gcd(denominator(bernfrac(n-1)),n)) \\ Charles R Greathouse IV, Jun 20 2011

a(n)=my(b=bernfrac(n-1));denominator(b)/denominator(b*n)/if(isprime(n),n,1) \\ Charles R Greathouse IV, Jun 20 2011

a(n)=if(isprime(n),1,my(b=bernfrac(n-1));denominator(b)/denominator(b*n)) \\ Charles R Greathouse IV, Jun 20 2011

a(n) = A166120(n)/ A050932(n-1).

A166333 The largest prime that divides A027642(n) (the denominator of the Bernoulli number B_n), or 1 if A027642(n) is 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 5, 1, 11, 1, 13, 1, 3, 1, 17, 1, 19, 1, 11, 1, 23, 1, 13, 1, 3, 1, 29, 1, 31, 1, 17, 1, 3, 1, 37, 1, 3, 1, 41, 1, 43, 1, 23, 1, 47, 1, 17, 1, 11, 1, 53, 1, 19, 1, 29, 1, 59, 1, 61, 1, 3, 1, 17, 1, 67, 1, 5, 1, 71, 1, 73, 1, 3, 1, 5, 1, 79, 1, 41, 1, 83, 1, 43, 1, 3, 1, 89, 1

Offset: 0

Paul Curtz, Oct 12 2009

nonn

The largest member of the extended prime list A008578 which divides the denominator of Bernoulli(n).

Essentially A073409 padded with 1's.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A006530, A027642, A073409, A089026, A166120.

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A166333(n) = A006530(denominator(bernfrac(n))); \\ Antti Karttunen, Dec 19 2018

a(n) = A006530(A027642(n)). - Antti Karttunen, Dec 19 2018

Edited and extended by R. J. Mathar, Oct 21 2009

Name and comment swapped by Antti Karttunen, Dec 19 2018

A356655 Clausen numbers based on the strictly proper divisors of n, 1 < d < n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 3, 1, 15, 1, 3, 1, 105, 1, 3, 1, 15, 1, 21, 1, 165, 1, 3, 1, 1365, 1, 3, 1, 15, 1, 231, 1, 255, 1, 3, 1, 25935, 1, 3, 1, 165, 1, 21, 1, 345, 1, 3, 1, 23205, 1, 33, 1, 15, 1, 399, 1, 435, 1, 3, 1, 465465, 1, 3, 1, 255, 1, 483, 1, 15, 1, 33, 1

Offset: 0

Peter Luschny, Aug 20 2022

nonn

Cf. A160014, A166120.

clausen := proc(n) numtheory[divisors](n) minus {1, n};
map(i -> i+1, %); select(isprime, %); mul(i, i=%)  end:
seq(clausen(n), n = 0..80);

a[n_] := Product[If[1 < d < n && PrimeQ[d + 1], d + 1, 1], {d, Divisors[n]}]; Array[a, 100, 0] (* Amiram Eldar, Aug 20 2022 *)

a(n) = if (n, vecprod(select(isprime, apply(x->x+1, setminus(divisors(n), [1,n])))), 1); \\ Michel Marcus, Aug 21 2022

a(n) = Product_{d | n} (d + 1), where d + 1 is prime and 1 < d < n.

cp's OEIS Frontend

A166123 If n is prime, a(n) = 1; otherwise, a(n) is gcd(n, d) where d is the denominator of the (n-1)-th Bernoulli number.

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A166333 The largest prime that divides A027642(n) (the denominator of the Bernoulli number B_n), or 1 if A027642(n) is 1.

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A356655 Clausen numbers based on the strictly proper divisors of n, 1 < d < n.

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PARI

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