A062169 -id:A062169 - cp's OEIS frontend

A062170 Maximum value of factorials mod n.

0, 1, 2, 2, 4, 2, 6, 6, 6, 6, 10, 6, 12, 10, 9, 8, 16, 12, 18, 6, 15, 16, 22, 6, 24, 24, 24, 24, 28, 24, 30, 24, 27, 32, 24, 24, 36, 36, 33, 24, 40, 36, 42, 32, 30, 44, 46, 24, 42, 40, 48, 48, 52, 36, 45, 48, 54, 52, 58, 24, 60, 60, 57, 56, 55, 60, 66, 64, 60, 50

Offset: 1

Henry Bottomley, Jun 11 2001

nonn

			a(15)=9 since factorials are 1, 2, 6, 24, 120, 720, etc. which mod 15 are 1, 2, 6, 9, 0, 0, etc. and the greatest value is 9.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A062169.

With[{fctrls=Factorial[Range[100]]}, Table[Max[Mod[Take[fctrls,n], n]], {n,100}]] (* Harvey P. Dale, Jan 03 2011 *)

A038203 Number of distinct values of factorials mod n.

Original entry on oeis.org

1, 2, 3, 3, 4, 3, 5, 4, 5, 5, 6, 4, 10, 6, 5, 5, 12, 5, 12, 5, 6, 7, 17, 4, 8, 11, 8, 7, 19, 5, 21, 6, 8, 13, 7, 6, 26, 13, 11, 5, 29, 6, 26, 8, 6, 18, 31, 5, 11, 8, 13, 12, 35, 8, 9, 7, 14, 20, 37, 5, 41, 22, 7, 8, 13, 8, 42, 14, 18, 7, 39, 6, 44, 27, 8, 15, 11, 11, 49, 6, 9, 30, 55, 7

Offset: 1

David W. Wilson

nonn

Assuming k! mod n is uniformly distributed mod n up to k = A002034(n), the first k! == 0 (mod n). This gives a(n) ~= (1-(1-1/n)^k)*n, which empirically appears to be a good estimate. For prime p, A002034(p) = p, so we would expect a(p) ~= (1-(1-1/p)^p)*p ~= (1-1/e)*p = 0.63212 p for large primes p. - David W. Wilson, Aug 01 2016

			a(15)=5 since factorials are 1, 2, 6, 24, 120, etc. which mod 15 are 1, 2, 6, 9, 0, etc. and so there are 5 distinct values.

David W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A038204, A062169.

nn=90;With[{frls=Range[nn]!},Table[Length[Union[Mod[#,n]&/@frls]],{n,nn}]] (* Harvey P. Dale, Oct 05 2011 *)

a(n)=my(t=1); #Set(vector(n,k,t=t*k%n)) \\ Charles R Greathouse IV, Aug 03 2016

A086330 a(n) = Sum_{m >= 2} m! mod n.

Original entry on oeis.org

0, 2, 4, 7, 2, 18, 8, 17, 12, 43, 8, 73, 32, 17, 24, 113, 26, 159, 12, 32, 76, 203, 8, 112, 164, 89, 60, 334, 32, 496, 88, 164, 232, 67, 44, 706, 292, 164, 32, 863, 74, 874, 164, 62, 456, 1097, 56, 291, 162, 317, 268, 1124, 116, 142, 88, 425, 566, 1560, 32, 2033, 930

Offset: 2

Walter Carlini, Aug 31 2003

A discrete infinite sum that has some rough analogies to the infinite series for exponentials.

			a(7) = 2! mod 7 + 3! mod 7 + 4! mod 7 + 5! mod 7 + 6! mod 7 + 7! mod 7 + 8! mod 7 + . . . = 2 mod 7 + 6 mod 7 + 24 mod 7 + 120 mod 7 + 720 mod 7 + 5040 mod 7 + 40320 mod 7 + ... = 2 + 6 + 3 + 1 + 6 + (all further values are zero) = 18.

Cf. A062169.

a(n) = sum(m=2, n, m! % n) \\ Michel Marcus, Jul 23 2013

def A086330(n):
    a, c = 0, 1
    for m in range(2,n):
        c = c*m%n
        if c==0:
            break
        a += c
    return a # Chai Wah Wu, Apr 16 2024

a(n) = -1 + Sum_{k=1..n} A062169(n, k). - Vladeta Jovovic, Sep 06 2003

Corrected and extended by Vladeta Jovovic, Sep 06 2003

A238532 Number of distinct factorial numbers congruent to -1 (mod n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 7, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

R. J. Mathar, Apr 02 2014

nonn

Number of solutions to k! == -1 (mod n), k>=1.
Counts the frequency of the value n-1 in the n-th row of triangle A062169.
Values 1..9 occur for the first time at n = 2, 7, 23, 59, 227, 401, 71, 3643, 62939, which are all prime numbers (see also A230315). Sequence A256519 gives composite k for which a(k) > 0. - Antti Karttunen, May 24 2021

			There are two 6's in the 7th row of A062169. Therefore a(7)=2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to factorial numbers

Cf. A000142, A049046, A062169, A230315, A256519.

A238532 := proc(n)
    local a,k ;
    a := 0 ;
    for k from 1 to n-1 do
        if modp(k!,n) = modp(-1,n) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Apr 02 2014

A238532(n) = { my(m=1,s=0); for(k=1,oo,m *= k; if(!(m%n),return(s), if(1+(m%n)==n, s++))); }; \\ Antti Karttunen, May 24 2021

A238532(n) = { my(m=Mod(1,n),s=0,x); for(k=1,oo, m *= Mod(k,n); x = lift(m); if(!x,return(s), if(x==(n-1), s++))); }; \\ (Much faster than above program) - Antti Karttunen, May 24 2021

A212320 Irregular triangle: T(n, k) = k! modulo prime(n), 1

Original entry on oeis.org

2, 2, 1, 4, 2, 6, 3, 1, 6, 2, 6, 2, 10, 5, 2, 5, 1, 10, 2, 6, 11, 3, 5, 9, 7, 11, 6, 1, 12, 2, 6, 7, 1, 6, 8, 13, 15, 14, 1, 12, 3, 8, 1, 16, 2, 6, 5, 6, 17, 5, 2, 18, 9, 4, 10, 16, 15, 16, 9, 1, 18, 2, 6, 1, 5, 7, 3, 1, 9, 21, 1, 12, 18, 22, 8, 13, 14, 22, 4

Offset: 2

Michel Marcus, Oct 25 2013

It is conjectured that only first and second row have all terms distinct.

This holds for n less than ten million. In Trudgian's terminology, there are no socialist primes less than 10^7. - Charles R Greathouse IV, Nov 05 2013

			Irregular triangle begins:
  2;
  2, 1, 4;
  2, 6, 3,  1, 6;
  2, 6, 2, 10, 5, 2, 5, 1, 10;

Vyacheslav M. Abramov, A solution to a Paul Erdos problem, arXiv:2504.19392 [math.NT], 2025.
W. D. Banks, F. Luca, I. E. Shparlinski, and H. Stichtenoth, On the Value Set of n! Modulo a Prime, Turk. J. Math., 29, (2005), 169-174.
B. Rokowska and A. Schinzel, Sur un problème de M. Erdős, Elem. Math., 15:84-85, 1960, MR117188 (22 #7970). [Broken link]
Tim Trudgian, There are no socialist primes less than 10^6, arXiv:1310.6403 [math.NT], 2013.

Cf. A062169.

row[n_] := With[{p = Prime[n]}, Mod[Range[2, p-1]!, p]]; Table[row[n], {n, 2, 9}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)

row(n) = {p = prime(n); for (i = 2, p-1, print1(i! % p, ", ");); print();}

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A062170 Maximum value of factorials mod n.

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A038203 Number of distinct values of factorials mod n.

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A086330 a(n) = Sum_{m >= 2} m! mod n.

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A238532 Number of distinct factorial numbers congruent to -1 (mod n).

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A212320 Irregular triangle: T(n, k) = k! modulo prime(n), 1

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