A060974 -id:A060974 - cp's OEIS frontend

A073277 Irregular primes with irregularity index two.

157, 353, 379, 467, 547, 587, 631, 673, 691, 809, 929, 1291, 1297, 1307, 1663, 1669, 1733, 1789, 1933, 1997, 2003, 2087, 2273, 2309, 2371, 2383, 2423, 2441, 2591, 2671, 2789, 2909, 2957, 3391, 3407, 3511, 3517, 3533, 3539, 3559, 3593, 3617, 3637, 3851

Offset: 1

Robert G. Wilson v, Jul 22 2002

nonn

Subsequence of A060974.

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, A091887, A000367, A060974, A060975 and A073276.

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 == 2, Print[p]], {n, 3, 550} ]

A060975 Irregular primes with irregularity index three.

Original entry on oeis.org

491, 617, 647, 1151, 1217, 1811, 1847, 2939, 3833, 4003, 4657, 4951, 6763, 7687, 8831, 9011, 10463, 10589, 12073, 13217, 14533, 14737, 14957, 15287, 15787, 15823, 16007, 17681, 17863, 18713, 18869, 20533, 20939, 24019, 24659, 25153, 26561

Offset: 1

Robert G. Wilson v, Jul 22 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated by Hart et al., terms 1..9824 from T. D. Noe, calculated by 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
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).

Cf. A000928, A000367, A060974, A060975, A073276 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 == 3, Print[p]], {n, 3, 1000} ]
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 == 3, Print@p], {n, 3, 13887} ]

Extended by Robert G. Wilson v, Sep 20 2006

A073276 Irregular primes (A000928) with irregularity index one.

Original entry on oeis.org

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} ]

A308765 Irregular triangle T(n,k) read by rows with 1 <= k <= A091887 even indices 2i such that n-th irregular prime p (A000928) divides the numerator of the Bernoulli numbers B_{2i} (A000367) with 0 <= 2i <= p-3.

Original entry on oeis.org

32, 44, 58, 68, 24, 22, 130, 62, 110, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 300, 100, 174, 200, 382, 126, 240, 366, 196, 130, 94, 194, 292, 336, 338, 400, 86, 270, 486, 222, 52, 90, 92, 22, 592, 522, 20, 174, 338, 428, 80, 226, 236, 242, 554, 48, 224, 408, 502, 628, 32, 12, 200, 378, 290, 514, 260, 732, 220, 330, 628, 544, 744, 102, 66, 868, 162, 418, 520, 820, 156, 166

Offset: 1

Martin Renner, Jun 23 2019

First index T(n,1) in row n is A035112(n).

			Triangle starts with
n = 1 => p = 37 divides the numerator of B_{32} = -7709321041217;
n = 2 => p = 59: B_{44};
n = 3 => p = 67: B_{58};
n = 4 => p = 101: B_{68};
n = 5 => p = 103: B_{24};
n = 6 => p = 131: B_{22};
n = 7 => p = 149: B_{130};
n = 8 => p = 157: B_{62}, B_{110};
n = 9 => p = 233: B_{84};
etc.

Cf. A000367, A000928, A035112, A060974, A060975, A073276, A073277, A091887.

T:=[]:
for j from 2 to 168 do
  p:=ithprime(j);
  B:=[]:
  for i from 1 to (p-3)/2 do
    if type(numer(bernoulli(2*i))/p,integer) then B:=[op(B),2*i]: fi:
  od:
  T:=[op(T),op(B)];
od:
op(T);

cp's OEIS Frontend

A073277 Irregular primes with irregularity index two.

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A060975 Irregular primes with irregularity index three.

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A073276 Irregular primes (A000928) with irregularity index one.

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A308765 Irregular triangle T(n,k) read by rows with 1 <= k <= A091887 even indices 2i such that n-th irregular prime p (A000928) divides the numerator of the Bernoulli numbers B_{2i} (A000367) with 0 <= 2i <= p-3.

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