A193267 -id:A193267 - cp's OEIS frontend

A300711 a(n) = A000367(n)/A001067(n).

1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 25, 13, 1, 7, 29, 1, 31, 1, 11, 17, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 5, 23, 47, 1, 49, 1, 17, 13, 53, 1, 5, 7, 19, 29, 59, 1, 61, 31, 1, 1, 65, 11, 67, 17, 23, 7, 71, 1, 73, 37

Offset: 1

Bernd C. Kellner, Mar 11 2018

nonn

a(n) is the trivial factor of the numerator of Bernoulli(2n) that divides 2n.
The remaining part of the (unsigned) numerator equals a product of powers of irregular primes, or 1 if and only if n = 1, 2, 3, 4, 5, 7.
Alternatively, a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing 2n and p-1 does not divide 2n.

			a(5) = 5, since Bernoulli(10) = 5/66 and Bernoulli(10)/10 = 1/132.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

A111008 equals the first entries and slightly differs, see a(35).

Cf. A000367, A001067, A193267, A195989, A300330, A002445, A075180.

using Nemo
function A300711(n)
    b = bernoulli(n)
    div(numerator(b), numerator(b*QQ(1,n)))
end
[A300711(n) for n in 2:2:148] |> println # Peter Luschny, Mar 11 2018

A300711 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
P := P*f[1]^f[2] fi od; P end: seq(A300711(n), n=1..74); # Peter Luschny, Mar 12 2018

Table[Numerator[BernoulliB[n]]/Numerator[BernoulliB[n]/n], {n, 2, 100, 2}]

a(n) = gcd(numerator(bernfrac(2*n)), 2*n) \\ Jianing Song, Apr 05 2021

upto(N)=bernvec(N);forstep(n=2,2*N,2,print1(gcd(numerator(bernfrac(n)), n),", ")) \\ Jeppe Stig Nielsen, Jun 22 2023

a(n) = numerator(Bernoulli(2n))/numerator(Bernoulli(2n)/(2n)).
a(n) * A195989(n) = n. - Peter Luschny, Mar 12 2018
From Jianing Song, Apr 05 2021: (Start)
a(n) = gcd(numerator(Bernoulli(2n)), 2n).
a(n) = A002445(n)*(2n)/A075180(2n-1). (End)

A195989 Quotient of denominators of (BernoulliB(2n)/n) and BernoulliB(2n).

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 2, 3, 16, 1, 18, 1, 20, 21, 2, 1, 24, 1, 2, 27, 4, 1, 30, 1, 32, 3, 2, 1, 36, 1, 2, 3, 40, 1, 42, 1, 4, 9, 2, 1, 48, 1, 50, 3, 4, 1, 54, 11, 8, 3, 2, 1, 60, 1, 2, 63, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 3, 4, 1, 78, 1, 80, 81, 2, 1, 84

Offset: 1

Paul Curtz, Dec 21 2012

nonn

The fixed points (entries equal to their index) are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 21, 24, 27, 30, 32, 36, 40, 42,... See A193267.
Are the indices of the 1's, that is 1, 5, 7, 11, 13,... , the sequence A069040 (checked to be true for their first 700 entries)? This provides another link between the Bernoulli numbers.
a(10*k) = 10, 20, 30, 40, 50, 60, 10, 70, 80, 90, 100,... for k= 1, 2, 3,....

			a(1) = 6/6 =1, a(2) = 60/30 =2, a(3) =126/42 =3, a(4) = 120/30 =4, a(5) = 66/66 =1.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

[Denominator(Bernoulli(2*n)/n)/Denominator(Bernoulli(2*n)): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018

A195989 := proc(n)
    q1 := denom(bernoulli(2*n)/n) ;
    q2 := denom(bernoulli(2*n)) ;
    q1/q2 ;
end proc: # R. J. Mathar, Jan 06 2013
# Alternatively, without Bernoulli numbers:
A195989 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
P := P*f[1]^f[2] fi od; n/P end: seq(A195989(n),n=1..84); # Peter Luschny, Mar 12 2018

a[n_] := Denominator[ BernoulliB[2*n]/n] / Denominator[ BernoulliB[2*n]]; Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Jan 04 2013 *)

a(n) = my(b=bernfrac(2*n)); denominator(b/n)/denominator(b); \\ Michel Marcus, Mar 12 2018

a(n) = A193267(2*n)/2 = A036283(n) / A002445(n).
a(n) = n/A300711(n). - Peter Luschny, Mar 12 2018
2a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing 2n and p-1 divides 2n. - Peter Luschny, Mar 12 2018

A300330 a(n) is the product over all prime powers p^e where p^e is the highest power of p dividing n and p-1 does not divide n.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 7, 1, 9, 5, 11, 1, 13, 7, 15, 1, 17, 1, 19, 1, 21, 11, 23, 1, 25, 13, 27, 7, 29, 5, 31, 1, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 23, 47, 1, 49, 25, 51, 13, 53, 1, 55, 7, 57, 29, 59, 1, 61, 31, 63, 1, 65, 11, 67, 17, 69, 35, 71, 1

Offset: 1

Peter Luschny, Mar 12 2018

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A193267, A195989, A300711.

using Nemo
function A300330(n) P = 1
    for (p, e) in factor(ZZ(n))
        ! divisible(ZZ(n), p - 1) && (P *= p^e) end
P end
[A300330(n) for n in 1:72] |> println

[n/(Denominator(Bernoulli(n)/n)/Denominator(Bernoulli(n))): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018

A300330 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(n)[2]; for f in F do if not divides(f[1]-1, n) then
P := P*f[1]^f[2] fi od; P end: seq(A300330(n), n=1..100);

a[n_]:=If[OddQ[n], 1, Denominator[BernoulliB[n]/n]/Denominator[BernoulliB[n]]]; Table[n/a[n], {n, 1, 100}] (* Vincenzo Librandi, Mar 12 2018 *)

a(n) * A193267(n) = n.

A354139 a(n) is the least positive integer m such that (k+1)^n + (k+2)^n + ... + (k+m)^n == 0 (mod n) for every positive integer k.

Original entry on oeis.org

1, 4, 3, 8, 5, 36, 7, 16, 3, 20, 11, 72, 13, 28, 15, 32, 17, 108, 19, 200, 21, 44, 23, 144, 5, 52, 3, 56, 29, 180, 31, 64, 33, 68, 35, 216, 37, 76, 39, 400, 41, 1764, 43, 88, 15, 92, 47, 288, 7, 20, 51, 104, 53, 324, 55, 112, 57, 116, 59, 1800, 61, 124, 21, 128, 65, 396, 67, 136, 69, 140, 71

Offset: 1

Dimitrios T. Tambakos, May 18 2022

nonn

a(n) divides n * A007947(n).

			a(2) = 4 because, for every positive integer k, (k+1)^2 + (k+2)^2 + (k+3)^2 + (k+4)^2 == 0 (mod 2), and no smaller positive integer satisfies this condition.

sum[n_, r_] := Mod[Sum[k^r, {k, 1, n}], r];
rad[r_] := Product[i[[1]], {i, FactorInteger[r]}];
seq[r_] := Table[sum[n, r], {n, 1, r*rad[r]}];
A354139[r_] := Piecewise[   {    {rad[r], OddQ[r]},
    {2*r, EvenQ[r] && PrimePowerQ[r]},
    {Length[FindRepeat[seq[r]]], EvenQ[r] && Not[PrimePowerQ[r]]}
    }
   ];
Table[A354139[r], {r, 1, 20}] (* Improved by Dimitrios T. Tambakos, Feb 08 2023 *)

isok(k, n) = my(p=sum(i=1, k, Mod(i+x, n)^n)); if (p==0, return(1)); for (i=1, n, if (subst(p, x, i) != 0, return(0))); return(1);
a(n) = my(k=1); while (!isok(k,n), k++); k; \\ Michel Marcus, May 21 2022

a(2^t) = 2^(t+1) for integers t>0.
a(n) = A007947(n) for odd integers n.
Conjecture: a(n) = A007947(n) * A193267(n).

cp's OEIS Frontend

A300711 a(n) = A000367(n)/A001067(n).

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A195989 Quotient of denominators of (BernoulliB(2n)/n) and BernoulliB(2n).

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A300330 a(n) is the product over all prime powers p^e where p^e is the highest power of p dividing n and p-1 does not divide n.

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A354139 a(n) is the least positive integer m such that (k+1)^n + (k+2)^n + ... + (k+m)^n == 0 (mod n) for every positive integer k.

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