

A051222


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


42



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
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OFFSET

1,1


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 StaudtClausen theorem.  N. J. A. Sloane, Jan 04 2004
The fraction of Bernoulli numbers with denominator 6 is roughly 1/6, see ErdősWagstaff. 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.
Numerator(B_{n}) mod Denominator(B_{n}) = 1. This relation stands also for B_{n} with denominator equal to 1, 2, 42 and 1806 (A014117).  Paolo P. Lava, Mar 30 2015
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


REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part IV, SpringerVerlag, 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.
Richard Sunseri, pAdic Lfunctions and densities relating to Bernoulli numbers, PhD thesis, University of Illinois, 1979.


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. 104112, MR 81c:10064.
K. L. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 7383.
R. Rado, A note on the Bernoullian numbers, J. London Math. Soc. 9 (1934) 8890.
Index entries for sequences related to Bernoulli numbers.


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 A051225A051230*) 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 *)


PROG

(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


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.
Sequence in context: A109255 A285990 A174594 * A194411 A017545 A280268
Adjacent sequences: A051219 A051220 A051221 * A051223 A051224 A051225


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Additional comments and references from Sam Wagstaff, Dec 20 2005


STATUS

approved



