A069040 -id:A069040 - cp's OEIS frontend

A070192 Numbers k such that k divides the numerator of B(2k) (the Bernoulli numbers), but gcd(3k, 8^k+1) > 3.

301, 737, 1505, 1655, 2107, 3197, 3311, 3913, 5117, 5159, 5219, 5719, 6275, 6923, 7385, 7513, 7525, 8107, 8275, 8729, 9331, 9581, 9835, 10535, 10849, 11137, 11585, 12341, 12529, 12943, 13301, 14003, 14147, 14749, 15953, 15985, 17759, 18361

Offset: 1

Benoit Cloitre and Dean Hickerson, Apr 26 2002

nonn

Equivalently, numbers that are in A069040 but not in A070191.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A027641, A069040, A070191, A070193.

testb[n_] := Select[First/@FactorInteger[n], Mod[2n, #-1]==0&]=={}; test8[n_] := GCD[3n, PowerMod[8, n, 3n]+1]==3; Select[Range[19000], testb[ # ]&&!test8[ # ]&]

isA070191(k) = gcd(3*k, Mod(8, 3*k)^k + 1) == 3;
isok(k) = {my(p = factor(k)[,1]); for(i = 1, #p, if(!((2*k) % (p[i]-1)), return(0))); !isA070191(k);} \\ Amiram Eldar, Apr 24 2025

A070193 Numbers k such that gcd(3k,8^k+1) = 3 but k does not divide the numerator of B(2k) (the Bernoulli numbers).

Original entry on oeis.org

253, 1081, 1771, 2485, 2783, 3289, 4301, 4807, 5405, 5819, 7337, 7567, 7843, 9361, 10373, 10879, 11891, 12397, 12425, 13409, 13861, 14053, 14927, 15433, 17395, 17963, 18145, 18377, 18469, 19481, 19987, 20539, 20999, 22517, 23023, 24541

Offset: 1

Benoit Cloitre and Dean Hickerson, Apr 26 2002

nonn

Equivalently, numbers is in A070191 but not in A069040.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A027641, A069040, A070191, A070192.

testb[n_] := Select[First/@FactorInteger[n], Mod[2n, #-1]==0&]=={}; test8[n_] := GCD[3n, PowerMod[8, n, 3n]+1]==3; Select[Range[25000], test8[ # ]&&!testb[ # ]&]

isA070191(k) = gcd(3*k, Mod(8, 3*k)^k + 1) == 3;
isok(k) = if(!isA070191(k), 0, my(p = factor(k)[,1]); for(i = 1, #p, if(!((2*k) % (p[i]-1)), return(1))); 0); \\ Amiram Eldar, Apr 24 2025

A070191 Numbers k such that gcd(3*k, 8^k+1) = 3.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 175, 179, 181

Offset: 1

Benoit Cloitre and Dean Hickerson, Apr 26 2002

nonn

The listed terms are the same as those in A069040, but the sequences are not identical. (The similarity is mostly explained by the absence of multiples of 2, 3 and 55 from both sequences.) See A070192 and A070193 for the differences.

The number of terms not exceeding 10^m, for m = 1, 2, ..., are 3, 32, 325, 3244, 32468, 324667, 3246642, 32466291, 324662816, 3246627133, ... . Apparently, the asymptotic density of this sequence exists and equals 0.32466... . - Amiram Eldar, Jun 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A069254.

Cf. A069040, A070192, A070193.

test8[n_] := GCD[3n, PowerMod[8, n, 3n]+1]==3; Select[Range[200], test8]

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

A281648 (Numerator of Bernoulli(2*n)) read mod n.

Original entry on oeis.org

0, 1, 1, 3, 0, 5, 0, 7, 1, 9, 0, 5, 0, 7, 5, 15, 0, 11, 0, 9, 1, 11, 0, 13, 0, 13, 19, 7, 0, 19, 0, 31, 11, 17, 0, 11, 0, 19, 13, 13, 0, 37, 0, 33, 35, 23, 0, 37, 0, 39, 34, 39, 0, 11, 5, 35, 19, 29, 0, 29, 0, 31, 61, 63, 0, 55, 0, 51, 23, 21, 0, 43, 0, 37, 50, 19

Offset: 1

Seiichi Manyama, Jan 26 2017

nonn

Conjecture: a(n) == n-1 (mod n) if only if n = 6, 10 or n = 2^k for k >= 0. This is true for n <= 1024. - Seiichi Manyama, Jan 27 2017

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000367, A060976, A069040, A070192, A070193, A281662.

f[n_] := Mod[Numerator[BernoulliB[2 n]], n]; Array[f, 77] (* Robert G. Wilson v, Jan 26 2017 *)

a(n)=numerator(bernfrac(2*n))%n \\ Charles R Greathouse IV, Jan 27 2017

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    ary << a[0]
  }
  ary
end
def A281648(n)
  a = bernoulli(2 * n)
  (1..n).map{|i| a[2 * i].numerator % i}
end

a(n) = A000367(n) mod n.

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A070192 Numbers k such that k divides the numerator of B(2k) (the Bernoulli numbers), but gcd(3k, 8^k+1) > 3.

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A070193 Numbers k such that gcd(3k,8^k+1) = 3 but k does not divide the numerator of B(2k) (the Bernoulli numbers).

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A070191 Numbers k such that gcd(3*k, 8^k+1) = 3.

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

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A281648 (Numerator of Bernoulli(2*n)) read mod n.

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