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

%I A051222 #96 May 04 2025 22:00:38
%S A051222 2,14,26,34,38,62,74,86,94,98,118,122,134,142,146,158,182,194,202,206,
%T A051222 214,218,254,266,274,278,298,302,314,326,334,338,362,386,394,398,422,
%U A051222 434,446,454,458,482,494,514,518,526,538,542,554,566,578
%N A051222 Numbers k such that Bernoulli number B_{k} has denominator 6.
%C A051222 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 A051222 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 A051222 Simon Plouffe reports (Feb 13 2007) that at B_{9083002} the proportion is 0.151848915149418661363281... and still decreasing very slowly.
%C A051222 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 A051222 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
%C A051222 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
%C A051222 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
%D A051222 B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
%D A051222 C. J. Moreno and S. S. Wagstaff, Sums of Squares of Integers, CRC Press, 2005, Sect. 3.9.
%D A051222 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 10.
%H A051222 T. D. Noe, <a href="/A051222/b051222.txt">Table of n, a(n) for n = 1..1000</a>
%H A051222 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 A051222 Richard R. Forberg, <a href="/A051222/a051222.pdf">A051222 and Exponential Expressions the Reproduce Certain Prime Subsets</a>, 2025.
%H A051222 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 A051222 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 A051222 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 A051222 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%t A051222 di[x_] := Divisors[x]
%t A051222 dp[x_] := Part[di[x], Flatten[Position[PrimeQ[1+di[x]], True]]]+1
%t A051222 Do[s=Length[dp[n]]; If[Equal[s, 2], Print[n]], {n, 1, 10000}] (* _Labos Elemer_ *)
%t A051222 Do[s=Denominator[BernoulliB[n]]; If[Equal[s, 6], Print[n]], {n, 1, 1000}] (* _Labos Elemer_ *)
%t A051222 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 A051222 Select[Range[600],Denominator[BernoulliB[#]]==6&] (* _Harvey P. Dale_, Dec 08 2011 *)
%o A051222 (PARI) for(n=1,10^3,if(denominator(bernfrac(n))==6,print1(n,", "))); \\ _Joerg Arndt_, Oct 28 2014
%o A051222 (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 A051222 Except for 2, all terms are even nontotient numbers. Proper subset of A005277: e.g., 50 and 90 are not here. - _Labos Elemer_
%Y A051222 A112772 is a subsequence. - _Bernd C. Kellner_, Mar 21 2018
%Y A051222 Cf. A045979, A000005, A067513, A002202, A005277.
%K A051222 nonn,nice,easy
%O A051222 1,1
%A A051222 _N. J. A. Sloane_
%E A051222 Additional comments and references from Sam Wagstaff, Dec 20 2005