A250406 -id:A250406 - cp's OEIS frontend

A250407 Near-Wilson primes (p = prime(n) satisfying (p-1)! == -1-A250406(n)*p (mod p^2)) with A250406(n) < 10.

2, 3, 5, 7, 13, 61, 71, 79, 157, 281, 563, 1277, 1777, 2339, 6311, 8233, 8543, 11047, 22907, 27689

Offset: 1

Felix Fröhlich, Nov 22 2014

A250406(n) is essentially A007619(n) modulo A000040(n) (see Crandall et al. (1997), p. 442).

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, Math. Comp., 83 (2014), 3071-3091.
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp., 66 (1997), 433-449.

forprime(p=1, 1e9, for(b=0, 9, if(Mod((p-1)!, p^2)==-1-b*p, print1(p, ", "); break({1}))))

A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

Original entry on oeis.org

1, 1, 1, 3, 5, 2, 8, 3, 14, 3, 18, 9, 9, 22, 18, 4, 18, 5, 1, 28, 30, 24, 3, 20, 46, 22, 47, 21, 15, 9, 57, 42, 15, 48, 28, 41, 48, 60, 85, 25, 74, 25, 52, 11, 32, 51, 17, 13, 34, 113, 13, 71, 2, 16, 64, 130, 81, 35, 37, 29, 39, 147, 68, 60, 71, 96, 92, 99, 12

Offset: 2

Felix Fröhlich, May 28 2015

nonn

p is in A001220 iff a(n) = 0. This is the case iff A014664(n) = A243905(n), which happens for n = 183 and n = 490.

Is a(n) = 0 for any other n, and, if yes, are there infinitely many such n?

Felix Fröhlich, Table of n, a(n) for n = 2..10000
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, 66 (1997), 433-449.

Cf. A001220, A195988, A241014, A244801, A246568, A250406, A250407, A306885, A338646, A343410.

a(n,p=prime(n))=abs(centerlift(Mod(2,p^2)^((p-1)/2))\/p)
apply(p->a(0,p), primes(100)[2..100]) \\ Charles R Greathouse IV, Jun 15 2015

a(n) = min(b(n) mod p, -b(n) mod p) where p = prime(n) and b(n) = Sum_{i=1..ceiling((p-1)/4)} (2i-1)^(p-2). - Daniel Chen, Sep 01 2022

A197636 Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

Original entry on oeis.org

2, 3, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 569

Offset: 1

Jonathan Sondow, Oct 19 2011

nonn

All primes except 5, 13, 563, and any other Wilson prime A007540 that may exist.
Same as primes p that do not divide their Wilson quotient ((p-1)!+1)/p.
Wilson's theorem says that (p-1)! == -1 (mod p) if and only if p is prime.
p = prime(i) is a term iff A250406(i) != 0. - Felix Fröhlich, Jan 24 2016
Complement of A007540 in A000040. - Felix Fröhlich, Jan 24 2016

			2 is a non-Wilson prime since (2-1)! = 1 ==/== -1 (mod 2^2).

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, Mathematics of Computation, 83 (2014), 3071-3091 (arXiv:1209.3436 [math.NT], 2012).
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Cf. A007540, A007619, A197633, A197634, A197635, A250406.

Select[Prime@ Range@ 104, Mod[Factorial[# - 1], #^2] != #^2 - 1 &] (* Michael De Vlieger, Jan 24 2016 *)

forprime(p=1, 500, if(Mod((p-1)!, p^2)!=-1, print1(p, ", "))) \\ Felix Fröhlich, Jan 24 2016

((p-1)!+1)/p =/= 0 (mod p), where p is prime.

A260209 Values A such that p=prime(n) satisfies binomial(2p-1, p-1) == 1 + A*p (mod p^4).

Original entry on oeis.org

1, 3, 25, 245, 121, 169, 867, 3249, 6877, 9251, 961, 15059, 57154, 61017, 68479, 106742, 201898, 208376, 107736, 176435, 330398, 237158, 158447, 213867, 903264, 856884, 21218, 755634, 1259386, 944906, 161290, 531991, 150152, 656914, 1287658, 592826, 640874

Offset: 1

Felix Fröhlich, Jul 19 2015

nonn

p is a Wolstenholme prime (A088164) iff a(n) == 0. This holds for n = 1944 and n = 157504.
When performing a search for Wolstenholme primes, one can choose an integer constant c >= 0 and record all primes p with A <= c in order to get a larger data set.
The values here asymptotically appear to grow more quickly than those in A260210.
It appears that a(n)/A260210(n) = A001248(n) for all n.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A088164, A241014, A244801, A250406, A258367, A260210.

f[n_] := Block[{p = Prime@ n}, (Mod[ Binomial[2p - 1, p - 1], p^4] - 1)/p]; Array[f, 37] (* Robert G. Wilson v, Jul 29 2015 *)

a(n) = p=prime(n); (lift(Mod(binomial(2*p-1, p-1), p^4))-1)/p

A260210 A034602(n) modulo prime(n).

Original entry on oeis.org

1, 5, 1, 1, 3, 9, 13, 11, 1, 11, 34, 33, 31, 38, 58, 56, 24, 35, 62, 38, 23, 27, 96, 84, 2, 66, 106, 74, 10, 31, 8, 34, 58, 26, 26, 144, 150, 140, 167, 137, 31, 107, 78, 157, 1, 103, 165, 97, 111, 60, 196, 48, 97, 259, 155, 175, 244, 13, 269, 34, 184, 222, 54

Offset: 3

Felix Fröhlich, Jul 19 2015

nonn

p is a Wolstenholme prime (A088164) iff a(n) = 0. This holds for n = 1944 and n = 157504.
When performing a search for Wolstenholme primes, one can choose an integer constant c >= 0 and record all primes with a(n) <= c in order to get a larger data set.
The values here appear to have a nicer asymptotic growth behavior than those in A260209.
It appears that A260209(n)/a(n) = A001248(n).
The formula only returns integers for primes greater than 3. - Robert G. Wilson v, Jul 29 2015

Felix Fröhlich, Table of n, a(n) for n = 3..10000

Cf. A034602, A088164, A241014, A244801, A250406, A258367, A260209.

f[n_] := Block[{p = Prime@ n}, (Mod[ Binomial[2p - 1, p - 1], p^4] - 1)/p^3]; Array[f, 60, 3] (* Robert G. Wilson v, Jul 29 2015 *)

a(n) = p=prime(n); lift(Mod(binomial(2*p-1, p-1)\p^3, p))

A034602(n)/prime(n) = A260209(n)/prime(n)^2, for n>2. - Robert G. Wilson v, Jul 29 2015

A338558 Absolute value q such that tau(p) == q (mod p), where p = prime(n) and tau(i) = A000594(i).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 7, 7, 1, 5, 10, 6, 11, 12, 20, 24, 14, 12, 3, 19, 6, 37, 20, 33, 20, 27, 50, 34, 36, 29, 18, 64, 4, 2, 66, 32, 3, 64, 61, 51, 60, 84, 95, 83, 63, 97, 42, 28, 61, 67, 32, 10, 29, 73, 37, 92, 16, 120, 31, 107, 120, 141, 145, 39, 12, 74, 150

Offset: 1

Felix Fröhlich, Dec 21 2020

nonn

These are essentially the values that can be used to define "near-misses" in a search of terms for A007659, similar to how "near-Wieferich primes", "near-Wilson primes" and "near-Wall-Sun-Sun primes" are defined in searches for Wieferich primes (A001220), Wilson primes (A007540) and Wall-Sun-Sun (Fibonacci-Wieferich) primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 (mod p), Journal of Integer Sequences 13 (2010), Article 10.7.4.

Cf. A001220, A007540, A007659, A295645.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme).

a[n_] := Module[{p = Prime[n]}, Min[Abs[Mod[RamanujanTau[p], {-p, p}]]]]; Array[a, 100] (* Amiram Eldar, Jan 10 2025 *)

a(n) = my(p=prime(n)); abs(centerlift(Mod(ramanujantau(p), p)))

a(n) = 0 iff prime(n) is a term of A007659.

A352858 a(n) = abs(E_{p-3} (mod p)) for p = prime(n), where E_i is the i-th Euler number (A000364).

Original entry on oeis.org

1, 2, 1, 3, 8, 7, 1, 3, 9, 4, 4, 4, 14, 7, 12, 16, 25, 22, 25, 4, 23, 33, 42, 15, 46, 18, 23, 38, 58, 2, 6, 55, 0, 37, 74, 63, 10, 61, 21, 38, 92, 89, 70, 79, 69, 59, 85, 22, 27, 69, 0, 45, 58, 96, 106, 6, 50, 28, 91, 133, 46, 147, 133, 38, 29, 128, 167, 116

Offset: 3

Felix Fröhlich, Apr 06 2022

nonn

a(n) = 0 iff p is a term of A198245.

These are the absolute values of the "A-values" that can be used to define "near-misses" in a search for terms of A198245 (cf. Mestrovic, 2014).

Romeo Mestrovic, A search for primes p such that Euler number E_p-3 is divisible by p, Mathematics of Computation 83 (2014), 2967-2976.

Cf. A000364, A198245.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme), A338558 (near-misses for A007659).

eulmod(n) = abs(centerlift(Mod(eulerfrac(n-3), n)))
a(n) = my(p=prime(n)); eulmod(p)

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A250407 Near-Wilson primes (p = prime(n) satisfying (p-1)! == -1-A250406(n)*p (mod p^2)) with A250406(n) < 10.

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A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

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A197636 Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

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A260209 Values A such that p=prime(n) satisfies binomial(2p-1, p-1) == 1 + A*p (mod p^4).

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A260210 A034602(n) modulo prime(n).

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A338558 Absolute value q such that tau(p) == q (mod p), where p = prime(n) and tau(i) = A000594(i).

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A352858 a(n) = abs(E_{p-3} (mod p)) for p = prime(n), where E_i is the i-th Euler number (A000364).

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