A051222 Numbers k such that Bernoulli number B_{k} has denominator 6.

2, 14, 26, 34, 38, 62, 74, 86, 94, 98, 118, 122, 134, 142, 146, 158, 182, 194, 202, 206, 214, 218, 254, 266, 274, 278, 298, 302, 314, 326, 334, 338, 362, 386, 394, 398, 422, 434, 446, 454, 458, 482, 494, 514, 518, 526, 538, 542, 554, 566, 578

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Alternative definition: let D(m) = set of divisors of m; sequence gives n such that the set 1 + D(n) contains only two primes, 2 and 3. E.g., n=98: D(98)={1,2,7,15,49,98}, 1+D = {2,3,8,16,50,99} of which only 2 terms are prime numbers: {2,3}. Observation by Labos Elemer, Jun 24 2002. This is a consequence of the von Staudt-Clausen theorem. - N. J. A. Sloane, Jan 04 2004

The fraction of Bernoulli numbers with denominator 6 is roughly 1/6, see Erdős-Wagstaff. But calculations by H. Cohen and G. Tenenbaum suggest that the fraction is closer to 1/7 (posting to Number Theory List around Dec 20 2005).

Simon Plouffe reports (Feb 13 2007) that at B_{9083002} the proportion is 0.151848915149418661363281... and still decreasing very slowly.

In his PhD thesis at the University of Illinois (see reference), Richard Sunseri proved that a higher proportion of Bernoulli denominators equal 6 than any other value.

Rado showed that for a given Bernoulli number B_n there exist infinitely many Bernoulli numbers B_m having the same denominator. As a special case, if n = 2p where p is an odd prime p == 1 (mod 3), then the denominator of the Bernoulli number B_n equals 6. - Bernd C. Kellner, Mar 21 2018

Conjecture: When the expression (p+q^b)/2 is required to be prime, p is prime, and q is a prime >=5, then all p values are prime congruent to 1 (mod 12) (A068228), if and only if the exponent b is a member of this set. - Richard R. Forberg, Apr 07 2025

There are additional exponential expressions conjectured for generating each of several known prime subsequences (e.g., Pythagorean primes, A002144) where the sequence is invariant to the exponent, if and only if the exponent is a member of this set. See Forberg link. - Richard R. Forberg, Apr 25 2025

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
• C. J. Moreno and S. S. Wagstaff, Sums of Squares of Integers, CRC Press, 2005, Sect. 3.9.
• H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 10.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Paul Erdős and Samuel S. Wagstaff, Jr., The fractional parts of the Bernoulli numbers, Illinois J. Math. 24 (1980), pp. 104-112, MR 81c:10064.
• Richard R. Forberg, A051222 and Exponential Expressions the Reproduce Certain Prime Subsets, 2025.
• K. L. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83.
• R. Rado, A note on the Bernoullian numbers, J. London Math. Soc. 9 (1934) 88-90.
• Richard Sunseri, Zeros of p-adic L-functions and densities relating to Bernoulli numbers, PhD thesis, University of Illinois, 1979.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Except for 2, all terms are even nontotient numbers. Proper subset of A005277: e.g., 50 and 90 are not here. - Labos Elemer

A112772 is a subsequence. - Bernd C. Kellner, Mar 21 2018

Cf. A045979, A000005, A067513, A002202, A005277.

Programs

Mathematica

di[x_] := Divisors[x]
dp[x_] := Part[di[x], Flatten[Position[PrimeQ[1+di[x]], True]]]+1
Do[s=Length[dp[n]]; If[Equal[s, 2], Print[n]], {n, 1, 10000}] (* Labos Elemer *)
Do[s=Denominator[BernoulliB[n]]; If[Equal[s, 6], Print[n]], {n, 1, 1000}] (* Labos Elemer *)
Do[s=1+Divisors[n];s1=Flatten[Position[PrimeQ[s], True]]; (*analogous [suitably modified] pairs of programs yield A051225-A051230*) s2=Part[s, s1];If[Equal[s2, {2, 3}], Print[n]], {n, 1, 100}] (* Labos Elemer *)
Select[Range[600],Denominator[BernoulliB[#]]==6&] (* Harvey P. Dale, Dec 08 2011 *)

PARI

for(n=1,10^3,if(denominator(bernfrac(n))==6,print1(n,", "))); \\ Joerg Arndt, Oct 28 2014

PARI

is(n)=if(n%2,return(0)); fordiv(n/2,d,if(isprime(2*d+1)&&d>1, return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2014

Extensions

Additional comments and references from Sam Wagstaff, Dec 20 2005