A070192 -id:A070192 - cp's OEIS frontend

A069040 Numbers k that divide the numerator of B(2k) (the Bernoulli numbers).

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, Apr 03 2002

nonn

Equivalently, k is relatively prime to the denominator of B(2k). Equivalently, there are no primes p such that p divides k and p-1 divides 2k. These equivalences follow from the von Staudt-Clausen and Sylvester-Lipschitz theorems.

The listed terms are the same as those in A070191, 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.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Seiichi Manyama)
I. Sh. Slavutskii, A note on Bernoulli numbers, Journal of Number Theory, Vol. 53, No. 2 (1995), pp. 309-310.

Cf. A027641, A070191, A070192, A070193.

A069040 := proc(n)
    option remember;
    if n=1 then
        1;
    else
        for k from procname(n-1)+1 do
            if numer(bernoulli(2*k)) mod k = 0 then
                return k;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jan 06 2013

testb[n_] := Select[First/@FactorInteger[n], Mod[2n, #-1]==0&]=={}; Select[Range[200], testb]

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

More information from Dean Hickerson, Apr 26 2002

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]

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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A069040 Numbers k that divide the numerator of B(2k) (the Bernoulli numbers).

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

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