A001067 -id:A001067 - cp's OEIS frontend

A057868 Denominator of "modified Bernoulli number" b(2n) = Bernoulli(2n)/(4n(2n)!).

48, 5760, 362880, 19353600, 958003200, 31384184832000, 2092278988800, 341459930972160000, 183927391818153984000, 32114306507931648000000, 620448401733239439360000, 81303558563123696133734400000, 9678995067038535254016000000, 2122022878497528469090467840000000

Offset: 1

Eric W. Weisstein

Note that Weisstein gives the formula b(n) = B(n)/(2*n*n!), and a(n) is the denominator of b(2*n). Numerators seem to be A141590 (not A001067 or A046968 or A255505). - Andrey Zabolotskiy, Dec 03 2022

			The sequence of modified Bernoulli numbers begins 1/48, -1/5760, 1/362880, -1/19353600, 1/958003200, -691/31384184832000, ...

D. Bar-Natan, T. T. Q. Le and D. P. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geometry and Topology 7-1 (2003) 1-31.
Eric Weisstein's World of Mathematics, Modified Bernoulli Numbers.
Index entries for sequences related to Bernoulli numbers.

Numerators seem to be A141590.

Cf. A001067.

seq(denom(bernoulli(2*n)/((4*n)*(2*n)!)), n = 1..14); # Peter Luschny, Dec 03 2022

a[n_] := Denominator[ BernoulliB[2n] / (8n^2*(2n-1)!)];
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Jun 07 2012 *)

Name edited by Andrey Zabolotskiy, Dec 03 2022

A085092 Numerators of constant term in a formal analog of Bernoulli numbers.

Original entry on oeis.org

1, 1, 1, 1, -432000, 1, -3456000, -9504000, -209520000, -389664000, -5952960000, -289595177265600

Offset: 2

N. J. A. Sloane, Aug 11 2003

			1, 1, 1, 1, -432000/691, 1, -3456000/3617, -9504000/43867, ...

M. Scarowsky, On a formal analogue of the Bernoulli numbers, J. Number Theory, 19 (1984), 228-232.

For denominators see A001067.

A090177 Numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n+1))).

Original entry on oeis.org

610, 1276, 1287, 1327, 1506, 1942, 1976, 2608, 2971, 3038, 3274, 3484, 3940, 4187, 4491, 4606, 4749, 4945, 5272, 5938, 6398, 6460, 6478, 6540, 6604, 6819, 7270, 7936, 8171, 8534, 8602, 8609, 8713, 9268, 9759, 9882, 9902, 9934, 10021

Offset: 1

mohammed bouayoun (bouyao(AT)wanadoo.fr), Feb 04 2004

nonn

Coincides with A090495(n) + A090496(n) - 1, except for order of terms.

Cf. A090179, A001067, A090495.

Do[ If[ Numerator[ BernoulliB[2n] / (2n)] != Numerator[ BernoulliB[2n] / (2n(2n + 1))], Print[n]], {n, 1, 10030}]

a(6)-a(39) from Robert G. Wilson v, Feb 09 2004

A090179 Ratio of numerator(Bernoulli(2n)/(2n)) to numerator(Bernoulli(2n)/(2n(2n+1))) for n's for which they are different.

Original entry on oeis.org

37, 37, 103, 59, 131, 37, 67, 37, 283, 59, 37, 101, 37, 67, 691, 37, 59, 157, 37, 37, 67, 59, 617, 103, 37, 593, 37, 37, 59, 101, 37, 67, 157, 37, 149, 59, 233, 37, 131

Offset: 1

mohammed bouayoun (bouyao(AT)wanadoo.fr), Feb 04 2004

nonn

Cf. A090177, A001067, A090495.

Do[ If[ r = Numerator[ BernoulliB[2n]/(2n)] / Numerator[ BernoulliB[2n]/(2n(2n + 1))]; r != 1, Print[r]], {n, 1, 10030}]

a(6)-a(39) from Robert G. Wilson v, Feb 09 2004

A090790 Numbers r arising in A090791.

Original entry on oeis.org

30, 42, 56, 66, 22, 20, 128, 60, 82, 162, 98, 82, 18

Offset: 1

Cino Hilliard, Feb 16 2004

These values of r correspond to the first 13 irregular primes produced by a/b.

			Given a, b as defined above and p=37, r=30, 52 = pk+r/2 = 37*1 + 30/2 is the smallest number that for a<>b a/b = 37.

Cf. A090495, A090496, A090791.

bern3(m,r) = { for(i=m,m, p=irprime(i); /* use the Somos script below to get irregular prime */ for(k=1,p, if(r%2,n=p*k+(p+r)/2,n=p*k+r/2); n2=n+n; a = numerator(bernfrac(n2)/(n2));
b = numerator(a/(n2-r)); v=a/b; if(a <> b && v==p,print(k","n","v);break) ) ) } /* A001067 */

irprime(n) = { my(p); if(n<1, 0, p=irprime(n-1) + (n==1); while(p = nextprime(p+2), forstep(i=2, p-3, 2, if( numerator(bernfrac(i))%p == 0, break(2)))); p) }; /* compute irregular primes irprime from - Michael Somos, Feb 04 2004 */

Given a = numerator(Bernoulli(2*n)/(2*n)) and b = numerator(a/(2*n-r)) for integer r positive or negative, then n>0 n = p*k+(p+r)/2 if r is odd and n = p*k+r/2 if r is even where k = 1, 2.. For every irregular prime p there is an r such that n is minimum.

A090791 Minimal numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n-r))) for some integer r.

Original entry on oeis.org

52, 80, 95, 134, 114, 141, 213, 187, 274, 338, 312, 312, 292

Offset: 1

Cino Hilliard, Feb 16 2004

These values of n correspond to the first 13 irregular primes produced by a/b.

			Given a,b as defined above and p=37,r=30, n=pk+r/2 = 37*k + 30/2 = 37k+15 = 52 = the smallest number that for a<>b a/b = 37.

Cf. A090790, A090495, A090496.

bern3(m,r) = { for(i=m,m, p=irprime(i); /* use the Somos script below to get irregular prime */ for(k=1,p, if(r%2,n=p*k+(p+r)/2,n=p*k+r/2); n2=n+n; a = numerator(bernfrac(n2)/(n2));
b = numerator(a/(n2-r)); v=a/b; if(a <> b && v==p,print(k","n","v);break) ) ) } /* A001067 */

irprime(n) = { my(p); if(n<1, 0, p = irprime(n-1) + (n==1); while(p = nextprime(p+2), forstep(i=2, p-3, 2, if( numerator(bernfrac(i))%p == 0, break(2)))); p) };  /* compute irregular primes irprime from - Michael Somos, Feb 04 2004 */

Given a = numerator(Bernoulli(2*n)/(2*n)) and b = numerator(a/(2*n-r)) for integer r positive or negative, then n>0 n = p*k+(p+r)/2 if r is odd and n = p*k+r/2 if r is even where k = 1, 2.. For every irregular prime p there is an r such that n is minimum.

A090798 Irregular primes in the ratio numerator(Bernoulli(2n)/(2n)) / numerator(Bernoulli(2n)/(2n(2n-r))) when these numerators are different and n is a minimum for some integer r. Duplication indicates irregularity index > 1.

Original entry on oeis.org

37, 59, 67, 101, 103, 131, 149, 157, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 353, 379, 379, 389, 401, 409, 421, 433, 461, 463, 467, 467, 491, 491, 491, 523, 541, 547, 547, 557, 577, 587, 587, 593, 607, 613, 617, 617, 617, 619, 631, 631, 647

Offset: 1

Cino Hilliard, Feb 16 2004

nonn

Only even values of r need to be tested.

See Table A.3, "Calculated irregular pairs of order 10 of primes below 1000," in B. C. Kellner.

Robert G. Wilson v, Table of n, a(n) for n = 1..2000
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

Cf. A090495 A090496.

f[p_] := Block[{c = 0, k = 1}, While[ 2k <= p - 3, If[ Mod[ Numerator@ BernoulliB[ 2k], p] == 0, c++]; k++]; c]; p = 5; lst = {}; While[p < 1001, AppendTo[lst, Table[p, {f@ p}]]; p = NextPrime@ p]; Flatten@ lst

\ prestore some ireg primes in iprime[] bernmin(m) = { for(x=1,m, p=iprime[x]; forstep(r=2,p,2, n=r/2+p; n2=n+n; a = numerator(bernfrac(n2)/(n2)); \ A001067 b = numerator(a/(n2-r)); \ if(a <> b,print(r","n","a/b)) if(a <> b,print1(a/b",")) ) ) }

Given a = numerator(Bernoulli(2*n)/(2*n)) and b = numerator(a/(2*n-r)) for integer r positive or negative, then n>0 n = p + r/2 For every irregular prime p there is an r such that n is minimum.

A281246 Least positive odd number m such that numerator of zeta(-m) are divisible by A000928(n).

Original entry on oeis.org

31, 43, 57, 67, 23, 21, 129, 61, 83, 163, 99, 83, 19, 155, 87, 291, 279, 185, 99, 199, 381, 125, 239, 365, 195, 129, 93, 291, 399, 85, 269, 221, 51, 89, 21, 591, 521, 19, 427, 79, 235, 47, 223, 407, 627, 31, 11, 377, 289, 513, 259, 731, 219, 329, 543, 743, 101

Offset: 1

Seiichi Manyama, Jan 18 2017

nonn

			zeta(-11) = 691/32760 and 691 are divisible by A000928(47). So a(47) = 11.

Seiichi Manyama, Table of n, a(n) for n = 1..150
Wikipedia, Herbrand-Ribet theorem

Cf. A000928, A001067.

A090793 Minimal numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n-r))) for some integer r and the first m irregular primes including irregularity index > 1.

Original entry on oeis.org

52, 80, 95, 134, 114, 141, 213, 187, 211, 274, 338, 312, 312, 292, 370, 350, 456, 486, 445, 502, 428, 465, 488, 591, 471, 540, 615, 558, 527, 513, 563, 636, 658, 659, 722, 583, 681, 789, 667, 602, 631, 632, 603, 902, 873, 626, 703, 785, 832, 670, 743, 764

Offset: 1

Cino Hilliard, Feb 16 2004

nonn

Only even values of r are tested.

Cf. A090495 A090496.

\ prestore some ireg primes in iprime[] bernmin(m) = { for(x=1,m, p=iprime[x]; forstep(r=2,p,2, n=r/2+p; n2=n+n; a = numerator(bernfrac(n2)/(n2)); \ A001067 b = numerator(a/(n2-r)); \ if(a <> b,print(r","n","a/b)) if(a <> b,print1(n",")) ) ) }

Given a = numerator(Bernoulli(2*n)/(2*n)) and b = numerator(a/(2*n-r)) for integer r positive or negative, then n>0 n = p + r/2 For every irregular prime p there is an r such that n is minimum.

A090800 r when numerator(Bernoulli(2n)/(2n)) and numerator(Bernoulli(2n)/(2n(2n-r))) are different and n is minimum for some integer r for the first i irregular primes. These include entries when the irregularity index > 1.

Original entry on oeis.org

30, 42, 56, 66, 22, 20, 128, 60, 108, 82, 162, 98, 82, 18, 154, 86, 290, 278, 184, 298, 98, 172, 198, 380, 124, 238, 364, 194, 128, 92, 192, 290, 334, 336, 398, 84, 268, 484, 220, 50, 88, 90, 20, 590, 520, 18, 172, 336, 426, 78, 224, 234, 240, 552, 46, 222, 406, 500

Offset: 2

Cino Hilliard, Feb 16 2004

nonn

This is a generalization of the concept in A090495 and A090496. One can change the code below from p = iprime[x] to p = prime(x) and see that data for only irregular primes is generated.

Cf. A090495 A090496.

\ prestore some ireg primes in iprime[] or use slower PARI BIF prime() bernmin(m) = { for(x=1,m, p=iprime[x]; forstep(r=2,p,2, n=r/2+p; n2=n+n; a = numerator(bernfrac(n2)/(n2)); \ A001067 b = numerator(a/(n2-r)); \ if(a <> b,print(r","n","a/b)) if(a <> b,print1(r",")) ) ) }

Given a = numerator(Bernoulli(2*n)/(2*n)) and b = numerator(a/(2*n-r)) for integer r positive or negative, then n>0 n = p + r/2. For every irregular prime p there is an r such that n is minimum.

cp's OEIS Frontend

A057868 Denominator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/(4*n*(2*n)!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A085092 Numerators of constant term in a formal analog of Bernoulli numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

A090177 Numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n+1))).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A090179 Ratio of numerator(Bernoulli(2*n)/(2*n)) to numerator(Bernoulli(2*n)/(2*n*(2*n+1))) for n's for which they are different.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A090790 Numbers r arising in A090791.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A090791 Minimal numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n-r))) for some integer r.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A090798 Irregular primes in the ratio numerator(Bernoulli(2*n)/(2*n)) / numerator(Bernoulli(2*n)/(2*n*(2*n-r))) when these numerators are different and n is a minimum for some integer r. Duplication indicates irregularity index > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A281246 Least positive odd number m such that numerator of zeta(-m) are divisible by A000928(n).

Views

Author

Keywords

Examples

Links

Crossrefs

A090793 Minimal numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n-r))) for some integer r and the first m irregular primes including irregularity index > 1.

Views

Author

A057868 Denominator of "modified Bernoulli number" b(2n) = Bernoulli(2n)/(4n(2n)!).

A090177 Numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n+1))).

A090179 Ratio of numerator(Bernoulli(2n)/(2n)) to numerator(Bernoulli(2n)/(2n(2n+1))) for n's for which they are different.

A090791 Minimal numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n-r))) for some integer r.

A090798 Irregular primes in the ratio numerator(Bernoulli(2n)/(2n)) / numerator(Bernoulli(2n)/(2n(2n-r))) when these numerators are different and n is a minimum for some integer r. Duplication indicates irregularity index > 1.

A090793 Minimal numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n-r))) for some integer r and the first m irregular primes including irregularity index > 1.

A090800 r when numerator(Bernoulli(2n)/(2n)) and numerator(Bernoulli(2n)/(2n(2n-r))) are different and n is minimum for some integer r for the first i irregular primes. These include entries when the irregularity index > 1.