A090495 Numbers k such that numerator(Bernoulli(2*k)/(2*k)) is different from numerator(Bernoulli(2*k)/(2*k*(2*k-1))).

574, 1185, 1240, 1269, 1376, 1906, 1910, 2572, 2689, 2980, 3238, 3384, 3801, 3904, 4121, 4570, 4691, 4789, 5236, 5862, 5902, 6227, 6332, 6402, 6438, 6568, 7234, 7900, 8113, 8434, 8543, 8557, 8566, 9232, 9611, 9670, 9824, 9891, 9898, 10564, 10587, 10754, 11230, 11247, 11535, 11691, 11896, 12562, 12965, 13019, 13228, 13246, 13355, 13484, 13894, 14560, 14714, 14957, 15176, 15226, 15346, 15892, 16558, 16668, 16944, 17035, 17224, 17387, 17890, 18379, 18406, 18534, 18556, 18761, 19222, 19598, 19888, 20090

Offset: 1

Author

N. J. A. Sloane, Feb 03 2004

Keywords

• nonn
• nice

Comments

Michael Somos (Feb 01 2004) discovered the remarkable fact that A001067 is different from A046968, even though they agree for the first 573 terms.

Numbers n such that A001067 is different from A046968, or alternatively, those n such that gcd(A001067(n),2n-1) is > 1.

If gcd(A000367(n), A000367(n+2)) <>1 then n = A090495(n) - (3*A090496(n) + 1)/2. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 08 2004

So far, all terms correspond to irregular primes. Notice that these numbers are generated by n=((2k+1)p+1)/2 where p is an irregular prime and k is some integer = 1,2,... . In the Excel spreadsheet provided at the link, you will notice that much larger firstborn irregular primes p tend to produce smaller values of k. E.g., p = 691, 683, 653, k = 5, 15, 23. So by some guessing we could test a given large irregular prime for the first few values of k. I found ip's 257, 293, 311 this way, but not the index. Also the spreadsheet shows the corresponding irregular primes where the Bacher forecast fails for firstborn irregular prime. - Cino Hilliard, Feb 15 2004

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..200
• Cino Hilliard, Bernoulli ratios [posted on Yahoo group B2LCC, Feb 04 2004]
• Eric Weisstein's World of Mathematics, Stirling's Series

Crossrefs

Cf. A090496, A001067, A046968, A092291. A274297.

Programs

Maple

a := n->numer(bernoulli(2*n)/(2*n)): b := n->numer(bernoulli(2*n)/(2*n*(2*n-1))): for n from 1 to 2000 do if a(n)<>b(n) then print(n,a(n)/b(n)); fi; od:

Mathematica

a[n_] := Numerator[BernoulliB[2n]/(2n)] (* A001067 *); b[n_] := Numerator[BernoulliB[2n]/(2n(2n-1))] (* A046968 *); For[n=1, n <= 580, n++, If[ a[n] != b[n], Print[n, " ", a[n]/b[n]] ] ]
k = 1; lst = {}; While[k < 38001, b = BernoulliB[2 k]; If [Numerator[b/(2 k)] != Numerator[b/(2 k (2 k - 1))], AppendTo[lst, k]; Print[{k}]]; k++ ]; lst (* Robert G. Wilson v, Aug 19 2010 *)

PARI

bern2(c,m1,m2) = { for(n=m1,m2, n2=n+n; a = numerator(bernfrac(n2)/(n2)); \ A001067 b = numerator(a/(n2-1)); if(a <> b,print("A("c") = "n","a/b);c++) ) } \\ Cino Hilliard

Extensions

a(1)-a(7) from Michael Somos and W. Edwin Clark, Feb 03 2004

a(8)-a(9) from Robert G. Wilson v, Feb 03 2004

a(10)-a(12) from Eric W. Weisstein, Feb 03 2004

a(13)-a(39) from Cino Hilliard, Feb 03 2004

a(40) from Eric W. Weisstein, Feb 04 2004

Many further terms from Cino Hilliard, Feb 15 2004