A038203 -id:A038203 - cp's OEIS frontend

A062169 Triangle T(n, k) = k! mod n for n >= 1, 1 <= k <= n.

0, 1, 0, 1, 2, 0, 1, 2, 2, 0, 1, 2, 1, 4, 0, 1, 2, 0, 0, 0, 0, 1, 2, 6, 3, 1, 6, 0, 1, 2, 6, 0, 0, 0, 0, 0, 1, 2, 6, 6, 3, 0, 0, 0, 0, 1, 2, 6, 4, 0, 0, 0, 0, 0, 0, 1, 2, 6, 2, 10, 5, 2, 5, 1, 10, 0, 1, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 11, 3, 5, 9, 7, 11, 6, 1, 12, 0, 1, 2, 6, 10, 8, 6, 0, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Jun 11 2001

			a(7,4) = 4! mod 7 = 24 mod 7 = 3. Rows are:
0;
1,0;
1,2,0;
1,2,2,0;
1,2,1,4,0;
1,2,0,0,0,0;
1,2,6,3,1,6,0;
1,2,6,0,0,0,0,0;
1,2,6,6,3,0,0,0,0;
1,2,6,4,0,0,0,0,0,0;

Harry J. Smith, Table of n, a(n) for n=1..1275

First zero in each row is when k=A002034. Maximum value in each row is A062170. Number of distinct values in each row is A038203. Cf. A000142, A048158, A051127.

Table[Mod[Range[n]!, n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)

{ n=0; for (k=1, 50, for (m=1, k, write("b062169.txt", n++, " ", m!%k)) ) } \\ Harry J. Smith, Aug 02 2009

Definition amended by Georg Fischer, Oct 27 2021

A049046 Number of k >= 1 with k! == 1 (mod n).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1

Offset: 1

David W. Wilson

nonn

The first occurrences for 0..10 are 1, 2, 5, 29, 17, 23, 199, 619, 3313, 4093, 3011, ... (see A049050). - Antti Karttunen, Oct 01 2018

			From _Antti Karttunen_, Oct 01 2018: (Start)
a(1) = 0 because 1 divides all factorial numbers (A000142): 1, 2, 6, 24, ... and thus there are no cases where the remainder would be 1.
a(3) = 1 as (1! mod 3) = 1, (2! mod 3) = 2 and for 3! and larger factorials the remainder is always 0. Thus there is exactly one case where the remainder is one.
a(5) = 2 as (1! mod 5) = 1, (2! mod 5) = 2, (3! mod 5) = 1, (4! mod 5) = 5, (5! mod 5) = 0, and so on ever after for larger factorials.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..12021

Cf. A000142, A002034, A071710, A038203, A049050.

Table[Length[Select[Range[100], Mod[#!, n] == 1 &]], {n, 1, 100}] (* G. C. Greubel, Oct 08 2018 *)

A049046(n) = { my(s=0, r, k=1); while((r=(k! % n))>0, s += (1==r); k++); (s); }; \\ Antti Karttunen, Oct 01 2018

Term a(1) corrected and the definition clarified by Antti Karttunen, Oct 01 2018

Definition further edited by Antti Karttunen, Oct 06 2018, based on feedback from David W. Wilson

A038204 Least k such that factorials have exactly n distinct residues mod k.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 22, 25, 55, 13, 26, 17, 34, 57, 76, 95, 23, 46, 29, 58, 31, 62, 93, 155, 319, 37, 74, 129, 41, 82, 47, 94, 141, 235, 53, 106, 59, 118, 71, 142, 61, 67, 134, 73, 146, 219, 365, 511, 79, 158, 237, 395, 553, 1241, 83, 166, 332, 747, 89, 178, 291, 485

Offset: 1

David W. Wilson

nonn

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

Cf. A038203.

s = Table[Length@ Union@ Mod[Range[10^3]!, n], {n, 2000}]; Table[FirstPosition[s, n], {n, 62}] // Flatten (* Michael De Vlieger, Aug 03 2016, Version 10 *)

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

cp's OEIS Frontend

A062169 Triangle T(n, k) = k! mod n for n >= 1, 1 <= k <= n.

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A049046 Number of k >= 1 with k! == 1 (mod n).

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A038204 Least k such that factorials have exactly n distinct residues mod k.

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