|
| |
|
|
A051222
|
|
Numbers n such that Bernoulli number B_{n} has denominator 6.
|
|
12
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
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}. Obervation by Labos E. (labos(AT)ana.sote.hu), Jun 24 2002. This is a consequence of the von Staudt-Clausen theorem. - N. J. A. Sloane (njas(AT)research.att.com), Jan 04, 2004
The fraction of Bernoulli numbers with denominator 6 is roughly 1/6, see Erdos-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 .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.
|
|
|
REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
P. Erdos and S. S. Wagstaff, The fractional parts of the Bernoulli numbers, Illinois J. Math. 24 (1980), pp. 104-112. MR 81c:10064.
K. L. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidsskrift F\"ur Mathematik, Afdeling B, vol. 28 (1915), pp. 73-83.
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, p-Adic L-functions 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
Index entries for sequences related to Bernoulli numbers.
|
|
|
MATHEMATICA
| ffi[x_] := Flatten[FactorInteger[x]] 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)
Program 1: Do[s=Denominator[BernoulliB[n]]; If[Equal[s, 6], Print[n]], {n, 1, 1000}] Program 2: 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)
Select[Range[600], Denominator[BernoulliB[#]]==6&] (* From Harvey P. Dale, Dec 08 2011 *)
|
|
|
CROSSREFS
| Except for 2, all terms are even nontotient numbers. Proper subset of A005277: e.g. 50, 90 are not here. (Labos)
Cf. A045979, A000005, A067513, A002202, A005277.
Sequence in context: A172048 A109255 A174594 * A194411 A017545 A109080
Adjacent sequences: A051219 A051220 A051221 * A051223 A051224 A051225
|
|
|
KEYWORD
| nonn,nice,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Additional comments and references from Sam Wagstaff, Dec 20 2005
|
| |
|
|
