A007540 -id:A007540 - cp's OEIS frontend

A277167 Prime numbers p such that (-1)^h + (h!)^2 == 0 (mod p^2) where h = (p-1)/2.

3, 11, 31, 47, 53

Offset: 1

René Gy, Oct 01 2016

The above congruence is true modulo p for all odd primes. See A089043. But like for Wilson congruence, it is true modulo p^2, for a restricted number of primes. After 53, the next one (if any) seems very far away (>500000).

The fact that the congruence is true modulo p for all odd primes was proved by Lagrange in 1771. Using a theorem of Mathews (1892) and Eisenstein's logarithmetic rule for the Fermat quotient, the condition stated in the definition can be restated as W_p == -2q_p(2) (mod p), where W_p is the Wilson quotient of p (A007619) and q_p(2) is the Fermat quotient of p, base 2 (A007663). - John Blythe Dobson, Jul 31 2017

			(-1)^((11-1)/2)+(((11-1)/2)!)^2 = 14399 = 7*11^2*17.

Lagrange, "Démonstration d’un théoreme nouveau concernant les nombres premiers," Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres [de Berlin], année 1771 (published 1783), 125-137.
G. B. Mathews, Theory of Numbers, part 1 [all published] (Cambridge, 1892), 318.

René Gy, Find the next "Wilson-like" prime.

Cf. A007540, A089043.

lista(nn) = forprime(p=3, nn, if ((((-1)^((p-1)/2)+(((p-1)/2)!)^2) % p^2) == 0, print1(p, ", "))); \\ Michel Marcus, Oct 02 2016

A282063 A(n, k) = k-th Wilson prime p of order n with p >= n and k running over the positive integers. Square array read by antidiagonals.

Original entry on oeis.org

5, 13, 2, 563, 3, 7

Offset: 1

Felix Fröhlich, Feb 05 2017

A Wilson prime of order n is a prime p such that (n-1)!*(p-n)!-(-1)^n == 0 (modulo p^2).

			Array A(n, k) starts:
      5,   13,  563
      2,    3,   11,  107, 4931
      7
  10429
      5,    7,   47
     11

Eric Weisstein's World of Mathematics, Wilson Prime
Wikipedia, Wilson prime

Cf. A007540 (row 1), A079853 (row 2), A152413 (row 17), A128666 (column 1).

is_wilson(n, order) = Mod((order-1)!*(n-order)!-(-1)^order, n^2)==0
table(rows, cols) = for(x=1, rows, my(i=0); forprime(p=x, , if(is_wilson(p, x), print1(p, ", "); i++; if(i==cols, print(""); break))))
table(4, 3) \\ print initial 4 rows and 3 columns of table

A290171 Numbers k such that (k-1)^2 < (k-1)! mod k^2.

Original entry on oeis.org

5, 13, 563, 1277, 780887

Offset: 1

Gionata Neri, Jul 23 2017

The Wilson primes (A007540) are terms of this sequence.

a(n) is prime or twice a prime. Otherwise (k-1)! mod k^2 = 0 for k > 9 where k is not a prime and not twice a prime. - David A. Corneth, Jul 23 2017

Cf. A007540.

Select[Range[10^4],(#-1)^2Giorgos Kalogeropoulos, Jul 23 2021 *)

for(n=1,1e5,a=(n-1)!%n^2;if((n-1)^2

is(n) = (n-1)^2 < lift(Mod((n-1)!, n^2)) \\ Felix Fröhlich, Jul 23 2017

val(n, p) = my(r=0); while(n, r+=n\=p); r
is(n) = my(f = factor(n), r = Mod(1, n^2)); if(#f~ > 2, return(0), if(#f~==2, if(f[1,1]!=2, return(0)))); forprime(p=2,n-1, r*=Mod(p,n^2)^val(n-1,p)); (n-1)^2 < lift(r) \\ David A. Corneth, Jul 23 2017

def ok(n):
    nn = n**2; f = 1%nn
    for k in range(1, n): f = f*k%nn
    return (n-1)**2 < f
print(list(filter(ok, range(1, 1300)))) # Michael S. Branicky, Jul 23 2021

# faster for initial segment of sequence
from math import factorial
def afind(limit, startk=1):
    k = startk; kkprev = (k-1)**2; f = factorial(k-1)
    while k < limit:
        kk = k*k
        if kkprev < f%kk: print(k, end=", ")
        kkprev = kk; f *= k; k += 1
afind(10000) # Michael S. Branicky, Jul 25 2021

a(5) from Chai Wah Wu, Jul 30 2017

A309398 a(n) is the nearest integer to log(log(10^n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Felix Fröhlich, Sep 27 2019

The sequence grows relatively slowly. For example, for n < 10^7, a(n) <= 17.

a(n) is roughly the expected number of Wieferich primes (cf. A001220 and Knauer, Richstein, 2005, p. 1560) as well as the expected number of Fibonacci-Wieferich primes (Wall-Sun-Sun primes) (cf. McIntosh, Roettger, 2007, p. 2091) and Wolstenholme primes (cf. A088164 and McIntosh, 1995, p. 387) with at most n digits. It is also roughly the expected number of Wilson primes with at most n digits (cf. A007540 and Costa, Gerbicz, Harvey, 2014).

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, arXiv:1209.3436 [math.NT], 2012; Mathematics of Computation, Vol. 83, No. 290 (2014), 3071-3091, DOI:10.1090/S0025-5718-2014-02800-7.
J. Knauer and J. Richstein, The continuing search for Wieferich primes, Mathematics of Computation, Vol. 74, No. 251 (2005), 1559-1563.
R. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica 71 (1995), 381-389.
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Mathematics of Computation, Vol. 76, No. 260 (2007), 2087-2094.

Cf. A000193, A001220, A007540, A088164.

Round[Log[Log[10^Range[90]]]] (* Harvey P. Dale, Jan 16 2024 *)

PARI
```
a(n) = round(log(log(10^n)))
```

a(n) = round(log(log(10^n))) = log n + O(1).

A319025 Primes p such that W_p == 2 (mod p), where W_p = A007619(n) and p = prime(n).

Original entry on oeis.org

19, 1187, 14296621, 16556218163369

Offset: 1

Felix Fröhlich, Sep 08 2018

These are the members of René Gy's set W_2 (cf. Gy, 2018).

The sequence is complete to 2*10^13, with the higher terms coming from a list of primes with small Wilson quotients in the article by Costa, Gerbicz, and Harvey. - John Blythe Dobson, Jan 05 2021

Edgar Costa, Robert Gerbicz, and David Harvey, A search for Wilson primes, Mathematics of Computation 83 (2014) 3071-3091.
R. Gy, Generalized Lerch Primes, Integers: Electronic Journal of Combinatorial Number Theory 18, (2018), #A10.

Cf. A007540, A197632.

forprime(p=1, , if(Mod(((p-1)!+1)/p, p)==2, print1(p, ", ")))

A365098 Primes p such that Sum_{k=1..p-1} q^2_p(k) == 0 (mod p), with q_p(k) a Fermat quotient.

Original entry on oeis.org

2, 11, 971

Offset: 1

Felix Fröhlich, Aug 21 2023

The congruence in the definition is given in Gy, 2018, Eq. 16.
The terms, except for the prime 2, satisfy the congruence B_{p-1} - 1 + 1/p == (B_{2p-2} - 1 + 1/p)/2 (mod p^2), with B_i a Bernoulli number (cf. Gy, 2018, Eq. 18).
Any odd prime that is a term of both A007540 and A197632, i.e., that is simultaneously a Wilson prime and a Lerch prime, is in this sequence (cf. Gy, 2018, Theorem 5).
An equivalent definition, better suited for computational purposes, is: "Primes p such that Sum_{k=1..p-1} (k^(p-1) - 1)^2 == 0 (mod p^3)." - John Blythe Dobson, Apr 30 2024
a(4) > 427000, if it exists (Gy, 2018). - Amiram Eldar, Aug 22 2023
a(4) > 39540000, if it exists. - John Blythe Dobson, Apr 30 2024

René Gy, Generalized Lerch primes, INTEGERS, 18 (2018), #A10.

Cf. A007540, A197632.

Join[{2}, Select[Prime[Range[2, 200]], Divisible[Numerator[BernoulliB[# - 1] - 1 + 1/# - (BernoulliB[2*# - 2] - 1 + 1/#)/2], #^2] &]] (* Amiram Eldar, Aug 22 2023 *)

forprime(p = 2, 10000, if(sum(j=1, p-1, (Mod(j, p^3)^(p-1) - 1)^2) % p^3 == 0, print1(p, ", "))) /* John Blythe Dobson, Apr 30 2024 */

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A277167 Prime numbers p such that (-1)^h + (h!)^2 == 0 (mod p^2) where h = (p-1)/2.

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A282063 A(n, k) = k-th Wilson prime p of order n with p >= n and k running over the positive integers. Square array read by antidiagonals.

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A290171 Numbers k such that (k-1)^2 < (k-1)! mod k^2.

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A309398 a(n) is the nearest integer to log(log(10^n)).

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A319025 Primes p such that W_p == 2 (mod p), where W_p = A007619(n) and p = prime(n).

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A365098 Primes p such that Sum_{k=1..p-1} q^2_p(k) == 0 (mod p), with q_p(k) a Fermat quotient.

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