A090495 -id:A090495 - cp's OEIS frontend

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.

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

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.

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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.

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Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

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.

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Comments

Crossrefs

Programs

PARI

Formula

A090800 r when numerator(Bernoulli(2*n)/(2*n)) and numerator(Bernoulli(2*n)/(2*n*(2*n-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.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

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.