A007540 -id:A007540 - cp's OEIS frontend

A283300 Primes p such that p^2 divides Bell(p) - 2.

2, 5, 11, 109, 509, 4099, 2951209

Offset: 1

Amiram Eldar, Mar 04 2017

A special case of Touchard's congruence is Bell(p) == 2 (mod p) for all primes p, where Bell(n) are the Bell numbers (A000110). These primes are for Touchard's congruence as Wieferich primes (A001220) are for Fermat's little theorem and Wilson primes (A007540) are for Wilson's theorem.

			For n=3, a(3)=11, Bell(11)=678570, Bell(11) - 2 = 11^2 * 61688.

J. Touchard, "Propriétés arithmétiques de certains nombres récurrents", Ann. Soc. Sci. Bruxelles A 53 (1933), pp. 21-31.

Eric Weisstein's World of Mathematics, Touchard's Congruence

Cf. A000110 (Bell numbers).

Select[Prime[Range[1000]], Divisible[BellB[#]-2, #^2] &]

a(7) from Hiroaki Yamanouchi, Aug 30 2018

A301316 a(n) = ((n-1)! + 1) mod n^2.

Original entry on oeis.org

0, 2, 3, 7, 0, 13, 35, 49, 64, 81, 11, 1, 0, 57, 1, 1, 85, 1, 38, 1, 1, 133, 184, 1, 1, 521, 1, 1, 522, 1, 589, 1, 1, 885, 1, 1, 259, 381, 1, 1, 656, 1, 559, 1, 1, 553, 282, 1, 1, 1, 1, 1, 1802, 1, 1, 1, 1, 2553, 1593, 1, 3416, 993, 1, 1, 1, 1, 804

Offset: 1

Stanislav Sykora, Mar 18 2018

nonn

By definition, when n > 1, a(n) = 0 then n is a Wilson prime (A007540).
For a(n) to equal 1, (n-1)! must be divisible by n^2 which is the prevailing case for large n. For example, all n which are a product of more than two distinct primes belong to this category. So do all proper powers of primes except 2^2, 2^3, and 3^2. Obviously, when a(n) = 1, then also A055976(n) = 1.
The cases of a(n) > 1 include, for example, all primes other than Wilson's and all numbers of the form n=2*p, where p is a prime.

			From _Muniru A Asiru_, Mar 20 2018: (Start)
((1-1)! + 1) mod 1^2 = (0! +1) mod 1 = 2 mod 1 = 0.
((2-1)! + 1) mod 2^2 = (1! +1) mod 4 = 2 mod 4 = 2.
((3-1)! + 1) mod 3^2 = (2! +1) mod 9 = 3 mod 9 = 3.
((4-1)! + 1) mod 4^2 = (3! +1) mod 16 = 7 mod 16 = 7.
((5-1)! + 1) mod 5^2 = (4! +1) mod 25 = 25 mod 25 = 0.
... (End)

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Wikipedia, Wilson prime

Cf. A000290, A038507, A007540, A055976, A301317.

List([1..60],n->(Factorial(n-1)+1) mod n^2); # Muniru A Asiru, Mar 20 2018

seq((factorial(n-1)+1) mod n^2,n=1..60); # Muniru A Asiru, Mar 20 2018

Array[Mod[(# - 1)! + 1, #^2] &, 67] (* Michael De Vlieger, Apr 21 2018 *)

a(n) = ((n-1)! + 1) % n^2; \\ Michel Marcus, Mar 18 2018

a(n) = ((n-1)! + 1) mod n^2. - Jon E. Schoenfield, Mar 18 2018

a(n) = A038507(n-1) mod A000290(n). - Michel Marcus, Mar 20 2018

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.

A377266 Primes p with the property that there exist nonnegative integers m,n such that m!*n! is congruent to either +1 or -1 mod p^2, with m + n = p - 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 47, 53, 59, 61, 71, 107, 137, 149, 173, 227, 251, 277, 313, 347, 349, 359, 367, 373, 409, 419, 443, 463, 467, 479, 491, 499, 521, 523, 541, 563, 577, 599, 607, 613, 617, 631, 643, 647, 677, 683, 739, 751, 757, 809, 811, 821, 823, 827, 829

Offset: 1

Richard Peterson, Oct 22 2024

nonn

The generalization of Wilson's Theorem that x!*(p-1-x)! is either +1 or -1 mod p when p is prime is known. This is investigated here for cases of p such that there is an x with x!*(p-1-x)! congruent to either +1 or -1 mod p^2.

			0!*4! + 1 = 5^2 and 4+0 = 5-1, so 5 is in the sequence.
1!*9! - 1 = 11^2*2999 and 1+9 = 10-1, so 11 is in the sequence.
0!*12! + 1 = 13^2*2834329 and 0+12 = 13-1, so 13 is in the sequence.
10!*6! + 1 = 17^2*83*108923 and 10+6=17-1, so 17 is in the sequence.
19!*39!-1 is divisible by 59^2 and 19+39=59-1, so 59 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..4011

A007540 is a subsequence.

Cf. A157249, A157250.

filter:= proc(p) local m,n,A,v;
  A:= Array(0..p);
  A[0]:= 1:
  for n from 1 to p do A[n]:= n*A[n-1] mod p^2 od:
  for m from 0 to (p-1)/2 do
    v:= A[m] * A[p-1-m] mod p^2;
    if v = 1 or v = p^2-1 then return true fi;
  od;
  false
end proc:
select(filter, [seq(ithprime(i),i=1..150)]); # Robert Israel, Dec 30 2024

isok(p)={if(isprime(p), for(m=0, p\2, my(t=(m!*(p-1-m)!%p^2)); if(t==1||t==p^2-1, return(1)))); 0} \\ Andrew Howroyd, Oct 22 2024

a(14) onwards from Andrew Howroyd, Oct 22 2024

A383578 Let p = prime(n), then a(n) is the p-smooth part of (p-1)!+1.

Original entry on oeis.org

2, 3, 25, 7, 11, 169, 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

Offset: 1

Mike Jones, Apr 30 2025

nonn

If x is an integer > 1 and p is a prime divisor of x, then a tower of x subordinate to p is an integer t such that there exists a prime divisor q of x such that q <= p, and t is the highest power of q that is a divisor of x.
If (p-1)!+1 = Product_{k} q_k^(e_k), then a(n) = Product_{k<=n} q_k^(e_k). - Sean A. Irvine, May 05 2025
Let p = prime(n). If m=p. Conjecture: a(n) = p^2 if n = 3, 6 or 103 and a(n) = p otherwise. - Chai Wah Wu, May 11 2025

			a(6) = 169 because the prime factorization of ((13 - 1)! + 1) is 13^2*2834329, and 13^2 = 169.

Cf. A007540, A060371, A383257.

a(n) = my(p=prime(n), x=(p-1)! + 1, f=factor((p-1)! + 1, nextprime(p+1))); for (i=1, #f~, if (f[i, 1] <= p, f[1, 1] = 1)); x/factorback(f); \\ Michel Marcus, Apr 30 2025

from sympy import prime, factorial
def A383578(n):
    p, c = prime(n), 1
    f = factorial(p-1)+1
    a, b = divmod(f,p)
    while not b:
        c *= p
        f = a
        a, b = divmod(f,p)
    return c # Chai Wah Wu, May 12 2025

a(n) = ((prime(n) - 1)! + 1) / A383257(n).

More terms from Michel Marcus, Apr 30 2025

A134295 a(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k).

Original entry on oeis.org

2, 2, 6, 16, 65, 312, 1813, 12288, 95617, 840960, 8254081, 89441280, 1060369921, 13649610240, 189550368001, 2824077312000, 44927447040001, 760034451456000, 13622700994560001, 257872110354432000, 5140559166898176001

Offset: 1

Alexander Adamchuk, Oct 17 2007

nonn

According to the Generalized Wilson-Lagrange Theorem, a prime p divides (p-k)!*(k-1)! - (-1)^k for all integers k > 0. p divides a(p) for prime p. Quotients a(p)/p are listed in A134296(n) = {1, 2, 13, 259, 750371, 81566917, 2642791002353, 716984262871579, 102688143363690674087, ...}. p^2 divides a(p) for prime p = {7, 71}.

Cf. A007540, A007619 (Wilson quotients: ((p-1)!+1)/p).

Cf. A134296 (quotients a(p)/p).

Table[ Sum[ (n-k)!*(k-1)! - (-1)^k, {k,1,n} ], {n,1,30} ]

a(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k).

A134296 Quotients A134295(p)/p = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k), where p = prime(n).

Original entry on oeis.org

1, 2, 13, 259, 750371, 81566917, 2642791002353, 716984262871579, 102688143363690674087, 21841112114495269555043222069, 17727866746681961093761724283871

Offset: 1

Alexander Adamchuk, Oct 17 2007

nonn

A134295(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k) = {2, 2, 6, 16, 65, 312, 1813, 12288, 95617, 840960, 8254081, ...}. According to the Generalized Wilson-Lagrange Theorem, a prime p divides (p-k)!*(k-1)! - (-1)^k for all integers k > 0. a(n) = A134295(p)/p for p = prime(n). a(n) is prime for n = {2, 3, 7, 9, 37, ...}. Corresponding prime terms in a(n) are {2, 13, 2642791002353, 102688143363690674087, ...}.

Cf. A007540, A007619 (Wilson quotients: ((p-1)!+1)/p).

Cf. A134295 (Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k)).

Table[ (Sum[ (Prime[n]-k)!*(k-1)! - (-1)^k, {k,1,Prime[n]} ]) / Prime[n], {n,1,20} ]

a(n) = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k) where p = prime(n).

A259230 a(n) = smallest k such that (A115091(n)-k)! == -1 (mod A115091(n)^2).

Original entry on oeis.org

1, 6, 1, 24, 64, 1, 384

Offset: 1

Felix Fröhlich, Nov 08 2015

The values of m in A115091.

A115091(n) is in A007540 iff a(n) = 1.

			a(2) = 6, because 6 is the smallest k such that (A115091(2)-k)! == -1 (mod A115091(2)^2), which yields the congruence (11-6)! == -1 (mod 11^2).

Cf. A007540, A115091.

t = Select[Prime@ Range@ 120, AnyTrue[Range@ #, Function[m, Divisible[m! + 1, #^2]]] &]; Table[k = 1; While[Mod[(t[[n]] - k)!, t[[n]]^2] != t[[n]]^2 - 1, k++]; k, {n, 7}] (* Michael De Vlieger, Nov 10 2015, Version 10 *)

forprime(p=1, , for(k=1, p-1, if(Mod((p-k)!, p^2)==-1, print1(k, ", "); break({1}))))

A275741 Sum of Wilson and Lerch remainders of n-th prime.

Original entry on oeis.org

1, 3, 10, 6, 6, 17, 15, 11, 25, 38, 9, 37, 47, 39, 86, 58, 107, 50, 101, 36, 98, 45, 123, 92, 170, 57, 80, 72, 158, 194, 194, 67, 78, 133, 120, 302, 144, 158, 128, 97, 91, 303, 76, 191, 139, 178, 302, 117, 242, 179, 335, 390, 362, 197, 290, 314, 327, 227, 429

Offset: 2

Felix Fröhlich, Aug 07 2016

nonn

a(n) = 0 if and only if prime(n) is in both A007540 and A197632, i.e., prime(n) is simultaneously a Wilson prime and a Lerch prime.
For n > 2, a(n) = 0 if and only if A027641(3*p-3) / A027642(3*p-3)-1 + 1/p == 0 (mod p^2), where p = prime(n) (cf. Dobson, 2016, theorem 2).
René Gy (see links) has shown that a number is simultaneously a Lerch prime and a Wilson prime if and only if it satisfies the congruence (p - 1)! + 1 == 0 (mod p^3). - John Blythe Dobson, Feb 23 2018

John Blythe Dobson, A Characterization of Wilson-Lerch Primes, Integers, 16 (2016), A51.
René Gy, Generalized Lerch Primes, Integers 18 (2018), A10.
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.

Cf. A002068, A007540, A197631, A197632.

a[n_] := Module[{p = Prime[n]}, Mod[((p-1)!+1)/p, p] + Mod[(Sum[(k^(p-1)-1)/p, {k, 1, p-1}] - ((p-1)!+1)/p)/p, p]];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Feb 15 2019 *)

a002068(n) = my(p=prime(n)); ((p-1)!+1)/p % p
a197631(n) = my(p=prime(n), m=p-1); sum(k=1, m, k^m, -p-m!)/p^2 % p
a(n) = a002068(n) + a197631(n)

a(n) = A002068(n) + A197631(n).

A283149 Largest k such that (p-1)! == -1 (mod p^k), where p = prime(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Felix Fröhlich, Mar 01 2017

nonn

a(n) > 1 iff A002068(n) = 0, i.e., iff p is a Wilson prime (A007540).

Is a(n) < 3 for all n?

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002068, A007540.

f:= proc(n) local p;
  p:= ithprime(n);
  padic:-ordp((p-1)!+1,p)
end proc:
map(f, [$1..200]); # Robert Israel, Apr 29 2021

Table[With[{p = Prime@ n}, SelectFirst[Reverse@ Range@ 10, Mod[(p - 1)!, #] == # - 1 &[p^#] &]], {n, 105}] (* Michael De Vlieger, Aug 20 2017 *)

a(n) = my(p=prime(n), k=1); while(Mod((p-1)!, p^k)==-1, k++); k-1

More terms from Antti Karttunen, Aug 20 2017

cp's OEIS Frontend

A283300 Primes p such that p^2 divides Bell(p) - 2.

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A301316 a(n) = ((n-1)! + 1) mod n^2.

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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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A377266 Primes p with the property that there exist nonnegative integers m,n such that m!*n! is congruent to either +1 or -1 mod p^2, with m + n = p - 1.

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A383578 Let p = prime(n), then a(n) is the p-smooth part of (p-1)!+1.

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A134295 a(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k).

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A134296 Quotients A134295(p)/p = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k), where p = prime(n).

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