A005451 -id:A005451 - cp's OEIS frontend

A088140 Duplicate of A005451 (for n >= 3).

2, 1, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63, 64, 65, 66, 1, 68, 69, 70, 1, 72, 1, 74, 75, 76, 77, 78, 1

Offset: 2

Roger L. Bagula, Nov 04 2003

dead

Previous name was: a(n) = 1 if n is an odd prime otherwise a(n) = n.

A135683 Duplicate of A005451.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63

Offset: 1

Mohammad K. Azarian, Dec 01 2007

dead

Previous name was: a(n) = 1 if n is a prime number, otherwise, a(n) = n.

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[IsPrime(n) select 1 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

seq(denom((1 + (n-1)!)/n), n=1..80); # G. C. Greubel, Nov 22 2022

Table[If[PrimeQ[n], 1, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)
a[n_] := ((n-1)! + 1)/n - Floor[(n-1)!/n] // Denominator; Table[a[n] , {n, 1, 63}] (* Jean-François Alcover, Jul 17 2013, after Minac's formula *)

def A135683(n):
    if n == 4: return n
    f = factorial(n-1)
    return 1/((f + 1)/n - f//n)
[A135683(n) for n in (1..63)]   # Peter Luschny, Oct 16 2013

a(n) = A088140(n), n >= 3. - R. J. Mathar, Oct 28 2008
a(n) = gcd(n, (n!*n!!)/n^2). - Lechoslaw Ratajczak, Mar 09 2019
a(n) = A005451(n), for n >= 2. - G. C. Greubel, Nov 22 2022

A007619 Wilson quotients: ((p-1)! + 1)/p where p is the n-th prime.

Original entry on oeis.org

1, 1, 5, 103, 329891, 36846277, 1230752346353, 336967037143579, 48869596859895986087, 10513391193507374500051862069, 8556543864909388988268015483871, 10053873697024357228864849950022572972973, 19900372762143847179161250477954046201756097561, 32674560877973951128910293168477013254334511627907

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

nonn

Suggested by the Wilson-Lagrange Theorem: An integer p > 1 is a prime if and only if (p-1)! == -1 (mod p).
Define b(n) = ((n-1)*(n^2 - 3*n + 1)*b(n-1) - (n-2)^3*b(n-2) )/(n*(n-3)); b(2) = b(3) = 1; sequence gives b(primes).
Subsequence of the generalized Wilson quotients A157249. - Jonathan Sondow, Mar 04 2016
a(n) is an integer because of to Wilson's theorem (Theorem 80, p. 68, the if part of Theorem 81, p. 69, given in Hardy and Wright). See the first comment. `This theorem is of course quite useless as a practical test for the primality of a given number n' ( op. cit., p. 69). - Wolfdieter Lang, Oct 26 2017

			The 4th prime is 7, so a(4) = (6! + 1)/7 = 103.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Oxford Science Publications, Clarendon Press, Oxford, 2003.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 277.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 234.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..100
Aminu Alhaji Ibrahim, Sa’idu Isah Abubaka, Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties, Advances in Pure Mathematics, 2016, 6, 409-419.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
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.
H. S. Wilf, Problem 10578, Amer. Math. Monthly, 104 (1997), 270.

Cf. A005450, A005451, A007540 (Wilson primes), A050299, A163212, A225672, A225906.
Cf. A261779.
Cf. A157249, A157250, A292691 (twin prime analog quotient).

Table[With[{p=Prime[n]},((p-1)!+1)/p],{n,15}] (* Harvey P. Dale, Oct 16 2011 *)

a(n)=my(p=prime(n)); ((p-1)!+1)/p \\ Charles R Greathouse IV, Apr 24 2015

a(n) = A157249(prime(n)). - Jonathan Sondow, Mar 04 2016

Definition clarified by Jonathan Sondow, Aug 05 2011

A005450 Numerator of (1 + Gamma(n))/n.

Original entry on oeis.org

2, 1, 1, 7, 5, 121, 103, 5041, 40321, 362881, 329891, 39916801, 36846277, 6227020801, 87178291201, 1307674368001, 1230752346353, 355687428096001, 336967037143579, 121645100408832001, 2432902008176640001

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..300
Achilleas Sinefakopoulos, Problem 10578 (Submitted solution.)
H. S. Wilf, Problem 10578, Amer. Math. Monthly, 104 (1997), 270.

Cf. A005451 (denominators).

[Numerator((1 + Factorial(n-1))/n): n in [1..30]]; // G. C. Greubel, Nov 21 2022

Table[Numerator[(1 + Gamma[n])/n], {n, 50}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Sep 12 2006 *)

[numerator((1+gamma(n))/n) for n in range(1,31)] # G. C. Greubel, Nov 21 2022

Define b(n) = ( (n-1)*(n^2-3*n+1)*b(n-1) - (n-2)^3*b(n-2) )/(n*(n-3)); b(2) = b(3) = 1; a(n) = numerator(b(n)), since b(n)=(1+Gamma(n))/n. - Joseph Biberstine (jrbibers(AT)indiana.edu), Sep 12 2006

Better description from Joseph Biberstine (jrbibers(AT)indiana.edu), Sep 12 2006

A364813 a(n) = Product_{k=2..n} k^ord(n, k) where ord(n, k) = 0 if k does not divide n, otherwise ord(n, k) = e where e is such that k^e divides n but k^(e + 1) does not.

Original entry on oeis.org

1, 2, 3, 16, 5, 36, 7, 256, 81, 100, 11, 3456, 13, 196, 225, 32768, 17, 17496, 19, 16000, 441, 484, 23, 1327104, 625, 676, 6561, 43904, 29, 810000, 31, 2097152, 1089, 1156, 1225, 362797056, 37, 1444, 1521, 10240000, 41, 3111696, 43, 170368, 273375, 2116, 47, 8153726976, 2401, 625000

Offset: 1

Michel Marcus, Oct 21 2023

nonn

a(n) is divisible by n and a(p) = p if p is prime. More general, if the base of the factors of the product is restricted to prime numbers then the positive integers are generated according to the fundamental theorem of arithmetic. - Peter Luschny, Apr 01 2025

Michel Marcus, Table of n, a(n) for n = 1..10000
Jeffrey C. Lagarias and Wijit Yangjit, The factorial function and generalizations, extended, arXiv:2310.12949 [math.NT], 2023. See Section 7.2 pp. 20-21 and Table 1 p. 29.
Wikipedia, p-adic valuation.

Cf. A363838 (also uses gamma), A000027, A005451, A381885 (a(n)/n).

with(padic): a := n -> local k; mul(k^ordp(n, k), k = 2.. n): seq(a(n), n = 1..50); # Peter Luschny, Apr 01 2025

Table[Product[k^IntegerExponent[n, k], {k, 2, n}], {n, 1, 50}] (* Peter Luschny, Apr 01 2025 *)

f(n, b) = sum(i=1, logint(n, b), n\b^i);
d(n,b) = f(n,b)-f(n-1,b);
a(n) = prod(b=2, n, b^d(n,b));

a(n) = Product_{b=2..n} b^d(n, b) where d(n, b) = gamma(n, b) - gamma(n-1, b) and gamma(n, b) = Sum_{i>=1} floor(n/b^i).

More explicit name from Peter Luschny, Apr 01 2025

A380441 Sum of the nonprimes dividing n and the number of distinct primes dividing n.

Original entry on oeis.org

1, 2, 2, 6, 2, 9, 2, 14, 11, 13, 2, 25, 2, 17, 18, 30, 2, 36, 2, 37, 24, 25, 2, 57, 27, 29, 38, 49, 2, 65, 2, 62, 36, 37, 38, 88, 2, 41, 42, 85, 2, 87, 2, 73, 72, 49, 2, 121, 51, 88, 54, 85, 2, 117, 58, 113, 60, 61, 2, 161, 2, 65, 96, 126, 68, 131, 2, 109, 72, 133, 2, 192, 2, 77, 118, 121, 80, 153, 2, 181, 119, 85, 2, 215, 88, 89, 90, 169, 2, 227, 94, 145, 96

Offset: 1

Wesley Ivan Hurt, Jun 21 2025

Inverse Möbius transform of A005451(n).

For each divisor d of n, add 1 if d is prime, else add d.

Cf. A000203 (sigma), A001221 (omega), A005171 (char nonprimes), A005451, A008472 (sopf), A023890.

Table[DivisorSigma[1, n] - Sum[p, {p, Select[Divisors[n], PrimeQ]}] + PrimeNu[n], {n, 100}]

a(n) = sigma(n) - sopf(n) + omega(n).
a(n) = Sum_{d|n} d^c(d), where c = A005171.
a(n) = Sum_{d|n} A005451(d).
a(p^k) = 1 - p + (p^(k+1)-1)/(p-1) for p prime, k >= 1. - Wesley Ivan Hurt, Jul 02 2025
a(n) = A023890(n) + A001221(n). - Wesley Ivan Hurt, Aug 31 2025

A381885 a(n) = Product_{k=2..n-1} k^ord(n, k) where ord(n, k) = 0 if k does not divide n, otherwise is the exponent of the highest power of k that divides n.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 32, 9, 10, 1, 288, 1, 14, 15, 2048, 1, 972, 1, 800, 21, 22, 1, 55296, 25, 26, 243, 1568, 1, 27000, 1, 65536, 33, 34, 35, 10077696, 1, 38, 39, 256000, 1, 74088, 1, 3872, 6075, 46, 1, 169869312, 49, 12500, 51, 5408, 1, 1417176, 55, 702464, 57

Offset: 1

Peter Luschny, Apr 01 2025

nonn

Wikipedia, p-adic valuation.

Cf. A364813, A005451.

with(padic): a := n -> local k; mul(k^ordp(n, k), k = 2.. n-1): seq(a(n), n = 1..57);

Table[Product[k^IntegerExponent[n, k], {k, 2, n - 1}], {n, 1, 57}]

a(n) = prod(k=2, n-1, k^valuation(n, k)); \\ Michel Marcus, Apr 01 2025

If the base of the factors of the product is restricted to prime numbers then A005451 is generated.
a(p) = 1 if p is prime.
a(n) = A364813(n) / n.

cp's OEIS Frontend

A088140 Duplicate of A005451 (for n >= 3).

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A135683 Duplicate of A005451.

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A007619 Wilson quotients: ((p-1)! + 1)/p where p is the n-th prime.

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A005450 Numerator of (1 + Gamma(n))/n.

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A364813 a(n) = Product_{k=2..n} k^ord(n, k) where ord(n, k) = 0 if k does not divide n, otherwise ord(n, k) = e where e is such that k^e divides n but k^(e + 1) does not.

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A380441 Sum of the nonprimes dividing n and the number of distinct primes dividing n.

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A381885 a(n) = Product_{k=2..n-1} k^ord(n, k) where ord(n, k) = 0 if k does not divide n, otherwise is the exponent of the highest power of k that divides n.

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