A051222 - OEIS

%I #79 Feb 19 2024 01:57:16

%S 2,14,26,34,38,62,74,86,94,98,118,122,134,142,146,158,182,194,202,206,

%T 214,218,254,266,274,278,298,302,314,326,334,338,362,386,394,398,422,

%U 434,446,454,458,482,494,514,518,526,538,542,554,566,578

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

%C 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

%C 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).

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

%C 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.

%C 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

%D B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.

%D C. J. Moreno and S. S. Wagstaff, Sums of Squares of Integers, CRC Press, 2005, Sect. 3.9.

%D H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 10.

%H T. D. Noe, <a href="/A051222/b051222.txt">Table of n, a(n) for n = 1..1000</a>

%H Paul Erdős and Samuel S. Wagstaff, Jr., <a href="http://www.renyi.hu/~p_erdos/1980-45.pdf">The fractional parts of the Bernoulli numbers</a>, Illinois J. Math. 24 (1980), pp. 104-112, MR 81c:10064.

%H K. L. Jensen, <a href="https://www.jstor.org/stable/24532219">Om talteoretiske Egenskaber ved de Bernoulliske Tal</a>, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83.

%H R. Rado, <a href="http://dx.doi.org/10.1112/jlms/s1-9.2.88">A note on the Bernoullian numbers</a>, J. London Math. Soc. 9 (1934) 88-90.

%H Richard Sunseri, <a href="http://hdl.handle.net/2142/68168">Zeros of p-adic L-functions and densities relating to Bernoulli numbers</a>, PhD thesis, University of Illinois, 1979.

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%t di[x_] := Divisors[x]

%t dp[x_] := Part[di[x], Flatten[Position[PrimeQ[1+di[x]], True]]]+1

%t Do[s=Length[dp[n]]; If[Equal[s, 2], Print[n]], {n, 1, 10000}] (* _Labos Elemer_ *)

%t Do[s=Denominator[BernoulliB[n]]; If[Equal[s, 6], Print[n]], {n, 1, 1000}] (* _Labos Elemer_ *)

%t 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_ *)

%t Select[Range[600],Denominator[BernoulliB[#]]==6&] (* _Harvey P. Dale_, Dec 08 2011 *)

%o (PARI) for(n=1,10^3,if(denominator(bernfrac(n))==6,print1(n,", "))); \\ _Joerg Arndt_, Oct 28 2014

%o (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

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

%Y A112772 is a subsequence. - _Bernd C. Kellner_, Mar 21 2018

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

%K nonn,nice,easy

%O 1,1

%A _N. J. A. Sloane_

%E Additional comments and references from Sam Wagstaff, Dec 20 2005