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Search: id:A090495
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| A090495 |
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Numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n-1))). |
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12
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Michael Somos discovered the remarkable fact that A001067 is different from A046968, even though they agree for the first 573 terms. Feb 01 2004
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 first born 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 first born irregular prime. - Cino Hilliard (hillcino368(AT)gmail.com), Feb 15 2004
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LINKS
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Eric Weisstein's World of Mathematics, Stirling's Series
Cino Hilliard, Bernoulli ratios.
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MAPLE
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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:
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MATHEMATICA
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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]] ] ]
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PROGRAM
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(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++) ) } (from C. Hilliard)
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CROSSREFS
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Cf. A090496, A001067, A046968, A092291.
Sequence in context: A142767 A076465 A049361 this_sequence A092291 A066154 A027456
Adjacent sequences: A090492 A090493 A090494 this_sequence A090496 A090497 A090498
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KEYWORD
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nonn,nice
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AUTHOR
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njas, Feb 03 2004
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EXTENSIONS
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a(1)-a(7) from Michael Somos and Edwin Clark, Feb 03, 2004.
a(8), a(9) from Robert G. Wilson v, Feb 03, 2004.
a(10)-a(12) from Eric Weisstein, Feb 03 2004
a(13)-a(39) from Cino Hilliard, Feb 03 2004
a(40) from Eric Weisstein, Feb 04 2004
Many further terms from Cino Hilliard, Feb 15 2004
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