A000928 -id:A000928 - cp's OEIS frontend

A073276 Irregular primes (A000928) with irregularity index one.

37, 59, 67, 101, 103, 131, 149, 233, 257, 263, 271, 283, 293, 307, 311, 347, 389, 401, 409, 421, 433, 461, 463, 523, 541, 557, 577, 593, 607, 613, 619, 653, 659, 677, 683, 727, 751, 757, 761, 773, 797, 811, 821, 827, 839, 877, 881, 887, 953, 971, 1061, 1091

Offset: 1

Robert G. Wilson v, Jul 22 2002

nonn

A prime p is regular if and only if the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) are not divisible by p.

In other words, irregular primes p dividing the numerator of B(2k) for a single k, 1<=k<(p-1)/2.

T. D. Noe, Table of n, a(n) for n=1..10000 (from Buhler et al.)
J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. A. Shokrollahi, Irregular Primes and Cyclotomic Invariants to 12 Million, J. Symbolic Computation 31, 2001, 89-96.
Bernoulli numbers, irregularity index of primes

Cf. A000928, A000367, A060974, A060975 and A073277.

Do[p = Prime[n]; k = 1; c = 0; While[ 2*k < p - 3, If[ Mod[ Numerator[ BernoulliB[2*k]], p] == 0, c++ ]; k++ ]; If[ c == 1, Print[p]], {n, 3, 200} ]

A091887 Irregularity index of the n-th irregular prime A000928(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1

Offset: 1

T. D. Noe, Feb 09 2004

nonn

See A091888 for definition.

T. D. Noe, Table of n, a(n) for n=1..10000 (from Buhler et al.)
J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. A. Shokrollahi, Irregular Primes and Cyclotomic Invariants to 12 Million, J. Symbolic Computation 31, 2001, 89-96.

Cf. A073277 (primes having irregularity index 2), A060975 (primes having irregularity index 3), A061576 (least prime having irregularity index n), A091888 (irregularity index of prime(n)).

tp=Table[p=Prime[i]; cnt=0; k=1; While[2k<=p-3, If[Mod[Numerator[BernoulliB[2k]], p]==0, cnt++ ]; k++ ]; cnt, {i, 400}]; Select[tp, #>0&]

A035112 Smallest even index 2a such that n-th irregular prime p (A000928(n)) divides Bernoulli_{2a} with 0<=2a<=p-3.

Original entry on oeis.org

32, 44, 58, 68, 24, 22, 130, 62, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 100, 200, 382, 126, 240, 366, 196, 130, 94, 292, 400, 86, 270, 222, 52, 90, 22, 592, 522, 20, 428, 80, 236, 48, 224, 408, 628, 32, 12, 378, 290, 514, 260, 732, 220, 330, 544, 744, 102

Offset: 1

N. J. A. Sloane

nonn

The ordered pair (p(n),a(n)) where p(n) is the n-th irregular prime is called an irregular pair. Some irregular primes, such as 157, are in more than one pair. See A091887 for the number of pairs for each irregular prime. See A092681 and A092682 for higher-order irregular pairs. - T. D. Noe, Mar 03 2004

			The first irregular prime (37) divides the numerator (-7709321041217) of the 32nd Bernoulli number.

L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350.

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A000928, A000367, A035112, A046753, A189683, A189684, A189685.

Do[ p = Prime[ n ]; k = 1; While[ 2*k < p - 3 && Mod[ Numerator[ BernoulliB[ 2*k ] ], p ] != 0, k++ ]; If[ 2*k != p - 3, Print[ 2*k ] ], { n, 3, 200} ]

More terms from Robert G. Wilson v, May 12 2001

A092681 Least number 2k such that p^2 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

Original entry on oeis.org

284, 914, 3292, 5768, 228, 12112, 11082, 6302, 11498, 40220, 34724, 51976, 1434, 74750, 67316, 21508, 63532, 39378, 67066, 64012, 91576, 137766, 137552, 105582, 158838, 147660, 175758, 194776, 173842, 102148, 132072, 107112, 127736, 29248

Offset: 1

T. D. Noe, Mar 03 2004

nonn

The ordered-pair (p(n),a(n)), where p(n) is the n-th irregular prime, is called a second-order irregular pair. Some irregular primes, such as 157, have more than one pair. See A091887 for a count of the pairs for each irregular prime.

Bernd Kellner, On irregular pairs of higher order (in German)

Cf. A000928, A035112, A091887, A092682.

A092291 Let p = n-th irregular prime, A000928(n). Then a(n) = smallest value of m such that numerator(Bernoulli(2m)/(2m)) / numerator(Bernoulli(2m)/(2m(2m-1))) equals p.

Original entry on oeis.org

574, 1269, 1910, 3384, 1185, 1376, 9611, 4789, 9670, 20946, 13019, 11247, 2689, 22708, 13355, 45251, 48407, 32653, 18761, 38706, 76391, 25563, 50310, 79023, 44948, 29864, 21716, 71441, 104339, 22993, 73572, 61549, 14714, 26122, 6227, 179369, 159687, 5862, 132157, 24925, 76023, 15346, 73479, 136956, 212240, 10587, 3801, 137040, 108520, 194171, 98550, 282532, 87272, 133081, 220187, 305002, 41764, 27268, 380180, 70921, 184940, 241076, 73858, 80108, 250927

Offset: 1

N. J. A. Sloane, based on a suggestion of Roland Bacher, Feb 05 2004

It was conjectured that a(n) = (1 + A000928(n) * (A035112(n) - 1))/2. However, Bernd Kellner's insightful paper shows that this formula first fails for the irregular prime 6449. - T. D. Noe, Feb 10 2004

Bernd Kellner, A conjecture about numerators of Bernoulli numbers

Term in A090495 corresponding to first occurrence of p in A090496.

(* This program is not convenient for a large number of terms *) irregularPrimeQ[p_] := Module[{k = 1}, While[2*k <= p-3 && Mod[ Numerator[ BernoulliB[2*k]], p] != 0, k++]; 2*k <= p-3]; irregularPrime[1] = 37; irregularPrime[n_] := irregularPrime[n] = Module[{p}, For[p = NextPrime[ irregularPrime[n-1]], True, p = NextPrime[p], If[ irregularPrimeQ[p], Return[p]]]]; a[n_] := a[n] = For[m = 1, True, m++, If[ Numerator[BernoulliB[2*m]/(2*m)] / Numerator[ BernoulliB[2*m]/(2*m*(2*m-1))] == irregularPrime[n], Return[m]]]; Table[ Print[a[n]]; a[n], {n, 1, 15}] (* Jean-François Alcover, Sep 27 2013 *)

Initial terms were computed by Roland Bacher, Feb 04 2004; further terms from Hans Havermann, Feb 05 2004 and T. D. Noe, Feb 06 2004

Offset modified by Jean-François Alcover, Sep 27 2013

A092682 Least number 2k such that p^3 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

Original entry on oeis.org

37580, 86464, 153640, 581468, 914250, 454892, 1510618, 3557642, 84974, 8905404, 11482532, 9629910, 1025814, 9252440, 6484016, 22003936, 17706562, 30054878, 18332698, 37340812, 39775150, 31082358, 5118308, 20315982, 57395934, 25079280

Offset: 1

T. D. Noe, Mar 03 2004

nonn

The ordered-pair (p(n),a(n)), where p(n) is the n-th irregular prime, is called a third-order irregular pair. Some irregular primes, such as 157, have more than one pair. See A091887 for a count of the pairs for each irregular prime.

Bernd Kellner, On irregular pairs of higher order (in German)

Cf. A000928, A035112, A091887, A092681.

A132360 Partial sum of irregular primes A000928.

Original entry on oeis.org

37, 96, 163, 264, 367, 498, 647, 804, 1037, 1294, 1557, 1828, 2111, 2404, 2711, 3022, 3369, 3722, 4101, 4490, 4891, 5300, 5721, 6154, 6615, 7078, 7545, 8036, 8559, 9100, 9647, 10204, 10781, 11368, 11961, 12568, 13181, 13798, 14417, 15048, 15695, 16348

Offset: 1

Jonathan Vos Post, Nov 08 2007

This sequence is infinite because Jensen proved in 1915 that there are infinitely many irregular primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000928, A172289.

More terms from R. J. Mathar, Nov 12 2007

A100639 Residues modulo 10 of the irregular primes (A000928).

Original entry on oeis.org

7, 9, 7, 1, 3, 1, 9, 7, 3, 7, 3, 1, 3, 3, 7, 1, 7, 3, 9, 9, 1, 9, 1, 3, 1, 3, 7, 1, 3, 1, 7, 7, 7, 7, 3, 7, 3, 7, 9, 1, 7, 3, 9, 3, 7, 3, 1, 7, 1, 7, 1, 3, 7, 9, 1, 1, 7, 9, 7, 1, 7, 9, 3, 1, 1, 1, 7, 9, 1, 3, 3, 1, 7, 9, 7, 9, 3, 1, 7, 1, 7, 9, 7, 7, 1, 9, 9, 9, 3, 9, 3, 9, 7, 9, 3, 9, 1, 7, 3, 9, 1, 3, 3, 9, 7

Offset: 1

Pahikkala Jussi, Dec 04 2004

			a(6) = 1 because the 6th irregular prime is 131 and 131 mod 10 = 1.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430 (but there are errors).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000928, A010879.

fQ[n_] := Block[{p = n, k = 1}, While[ 2*k <= p - 3 && Mod[ Numerator[ BernoulliB[ 2*k ]], p ] != 0, k++ ]; 2k != p - 1]; Mod[ Select[ Prime[ Range[2, 275]], fQ[ # ] &], 10] (* Robert G. Wilson v, Dec 10 2004 *)

a(n) = A010879(A000928(n)). - Amiram Eldar, Jul 02 2024

More terms from Robert G. Wilson v, Dec 10 2004

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.

A092901 Number of irregular primes (A000928) less than 10^n.

Original entry on oeis.org

0, 3, 64, 497, 3789, 30870, 261871, 2266482, 20006269

Offset: 1

Eric W. Weisstein, Mar 13 2004

Buhler maintains a list of regular and irregular primes. Data about each irregular prime includes the index of irregularity, irregular pairs, Vandiver residue and two cyclotomic residues.

Corrected using latest data from Buhler and Harvey. - T. D. Noe, Apr 27 2011

			The smallest irregular prime is 37.

J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. A. Shokrollahi, Irregular Primes and Cyclotomic Invariants to 12 Million, J. Symbolic Computation 31, 2001, 89-96.
Joe Buhler and David Harvey, Irregular primes to 163 million, arXiv:0912.2121 [math.NT], 2009.
Joe Buhler and David Harvey, Irregular primes to 163 million
William Hart, David Harvey and Wilson Ong, Irregular primes to two billion, Mathematics of Computation, Vol. 86, No. 308 (2017), pp. 3031-3049; also available at arXiv:1605.02398 [math.NT], 2016.
David Harvey, Irregular primes to two billion (includes a list of all primes less than 2^31).
Eric Weisstein's World of Mathematics, Irregular Prime

Cf. A000928.

a(6) and a(7) from T. D. Noe, Mar 31 2004
a(8) from T. D. Noe, Apr 27 2011
a(9) from Amiram Eldar, Mar 05 2019

cp's OEIS Frontend

A073276 Irregular primes (A000928) with irregularity index one.

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A091887 Irregularity index of the n-th irregular prime A000928(n).

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A035112 Smallest even index 2a such that n-th irregular prime p (A000928(n)) divides Bernoulli_{2a} with 0<=2a<=p-3.

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A092681 Least number 2k such that p^2 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

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A092291 Let p = n-th irregular prime, A000928(n). Then a(n) = smallest value of m such that numerator(Bernoulli(2*m)/(2*m)) / numerator(Bernoulli(2*m)/(2*m*(2*m-1))) equals p.

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A092682 Least number 2k such that p^3 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

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A132360 Partial sum of irregular primes A000928.

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A100639 Residues modulo 10 of the irregular primes (A000928).

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

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A092291 Let p = n-th irregular prime, A000928(n). Then a(n) = smallest value of m such that numerator(Bernoulli(2m)/(2m)) / numerator(Bernoulli(2m)/(2m(2m-1))) equals p.