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

Author

Benoit Cloitre, Apr 03 2002

Keywords

• nonn

Comments

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.

References

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

Links

• 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.

Crossrefs

Cf. A027641, A070191, A070192, A070193.

Programs

Maple

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

Mathematica

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

PARI

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

Extensions

More information from Dean Hickerson, Apr 26 2002