A060371 -id:A060371 - cp's OEIS frontend

A254924 a(n) = (A060371(n) - A094998(n))/A056604(n) for n > 1, with a(1)=1.

1, 0, 0, 1, 130, 1329, 1707670, 27502484, 209927657739, 130904517147542068, 3673771932850374193, 69623451054783204822486486, 3724616892817543661693877073170, 149157913707716515940392007441860, 12429106799179771738076359013310638297

Offset: 1

Bruno Berselli, Feb 12 2015 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

Let theta(p) be the smallest nonnegative solution z to the system of congruences z == 0 (mod p), z == 1 (mod v(p-1)), where p is a prime and v(p-1) = lcm(1,...,p-1). Theta(p) is unique mod lcm(p, v(p-1)), therefore it is unique mod v(p). Since both (p-1)!+1 and theta(p) are solutions to these congruences, ((p-1)!+1 - theta(p))/v(p) is always an integer. The sequence lists the values of this ratio (assuming theta(2)=0 and p=prime(n)).

			For n=5, a(5) = (A060371(5) - A094998(5))/A056604(5) = (3628801 - 25201)/27720 = 130.

Bruno Berselli, Table of n, a(n) for n = 1..50
Umberto Cerruti, Il Teorema Cinese dei Resti (in Italian), 2015. The sequence is on page 21.

Cf. A000040, A056604, A060371, A094998, A254939, A255010.

[(Factorial(p-1)+1-Modinv(p,Lcm([1..p-1]))*p)/Lcm([1..p]): p in PrimesUpTo(50)];

with(numtheory): P:=proc(q)  local a,j,k,ok,n;  print(1); a:=[1];
for n from 3 to q do k:=0; a:=[op(a),n]; if isprime(n) then ok:=0;  while ok=0 do ok:=1;
k:=k+1; for j from 2 to n-1 do if not (k*n mod j)=1 then ok:=0; break; fi; od; od;
print((((n-1)!+1)-k*n)/lcm(op(a))); fi; od; end: P(100); # Paolo P. Lava, Feb 16 2015

r[k_] := LCM @@ Range[k]; s[k_] := PowerMod[k, -1, r[k - 1]] k; w[k_] := ((k - 1)! + 1 - s[k])/r[k]; Table[w[Prime[n]], {n, 1, 20}]

A383257 Let p = prime(n), then a(n) is the non-p-smooth part of (p-1)!+1.

Original entry on oeis.org

1, 1, 1, 103, 329891, 2834329, 1230752346353, 336967037143579, 48869596859895986087, 10513391193507374500051862069, 8556543864909388988268015483871, 10053873697024357228864849950022572972973, 19900372762143847179161250477954046201756097561, 32674560877973951128910293168477013254334511627907

Offset: 1

Mike Jones, Apr 29 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) and k = (p-1)!+1. If mChai Wah Wu, May 11 2025

			a(6) = 2834329 because ((13 - 1)! + 1)/w = (12! + 1)/w = (13^2*2834329)/w = 2834329, where w is the product of the towers of ((13 - 1)! + 1) subordinate to 13, w equaling 13^2.

Cf. A060371, A383578.

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

from sympy import prime, factorial
def A383257(n):
    p = prime(n)
    f = factorial(p-1)+1
    a, b = divmod(f,p)
    while not b:
        f = a
        a, b = divmod(f,p)
    return f # Chai Wah Wu, May 12 2025

More terms from Michel Marcus, Apr 30 2025

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

A049563 a(n) = ((prime(n)-1)! + 1) mod (prime(n) + 2).

Original entry on oeis.org

2, 3, 4, 1, 7, 1, 10, 1, 1, 16, 1, 1, 22, 1, 1, 1, 31, 1, 1, 37, 1, 1, 1, 1, 1, 52, 1, 55, 1, 1, 1, 1, 70, 1, 76, 1, 1, 1, 1, 1, 91, 1, 97, 1, 100, 1, 1, 1, 115, 1, 1, 121, 1, 1, 1, 1, 136, 1, 1, 142, 1, 1, 1, 157, 1, 1, 1, 1, 175, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 211, 1, 217, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer

nonn

Residue of (prime(n)-1)!+1 modulo prime(n)+2.

			a(3) = 4 since prime(3) = 5, and 4! + 1 = 25 gives residue 4 when divided by prime(3) + 2 = 7.

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

Cf. A052147, A060371.

[(Factorial(p-1)+1) mod (p+2): p in PrimesUpTo(500)]; // Bruno Berselli, Apr 10 2015

Table[Mod[(Prime[k] - 1)! + 1, Prime[k] + 2], {k, 1, 200}]

a(n) = ((prime(n)-1)! + 1) % (prime(n) + 2); \\ Michel Marcus, May 28 2018

[Mod(factorial(p-1)+1,p+2) for p in primes(500)] # Bruno Berselli, Apr 10 2015

a(n) = A060371(n) mod A052147(n). - Amiram Eldar, Mar 13 2025

A062411 a(n) = (-1)^(p-1)*(p-1)! + 1 where p = prime(n).

Original entry on oeis.org

0, 3, 25, 721, 3628801, 479001601, 20922789888001, 6402373705728001, 1124000727777607680001, 304888344611713860501504000001, 265252859812191058636308480000001

Offset: 1

Jason Earls, Jul 09 2001

Apart from the first term, the same as A060371. - R. J. Mathar, Oct 02 2008

D. M. Burton, Elementary Number Theory, Allyn and Bacon, Inc. Boston, MA, 1976, p. 164.

for(n=1,13,print((-1)^(prime(n)-1)*(prime(n)-1)!+1))

cp's OEIS Frontend

A254924 a(n) = (A060371(n) - A094998(n))/A056604(n) for n > 1, with a(1)=1.

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A383257 Let p = prime(n), then a(n) is the non-p-smooth part of (p-1)!+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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Python

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A049563 a(n) = ((prime(n)-1)! + 1) mod (prime(n) + 2).

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Magma

Mathematica

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A062411 a(n) = (-1)^(p-1)*(p-1)! + 1 where p = prime(n).

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