A062170 -id:A062170 - 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

A221861 The least number k that maximizes k! mod n.

Original entry on oeis.org

0, 0, 2, 2, 4, 2, 3, 3, 3, 3, 5, 3, 12, 4, 4, 4, 16, 5, 9, 3, 5, 6, 14, 3, 4, 4, 4, 4, 18, 4, 30, 4, 6, 9, 4, 4, 36, 6, 8, 4, 40, 5, 21, 5, 5, 10, 23, 4, 7, 7, 10, 7, 52, 8, 9, 6, 13, 7, 15, 4, 8, 14, 5, 5, 5, 6, 18, 8, 17, 5, 7, 5, 72, 21, 5, 14, 9, 8, 23, 5

Offset: 1

Aaron Weiner, Apr 10 2013

nonn

			For n=11, we see that the factorial of 5 (120), modulo 11 is 10, which is the highest possible value, so the 11th term is 5.

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

Cf. A062170.

(1..100).map{|n|(0..n).max_by{|x|[(1..x).inject(1,:*)%n,-x]}}

A327649 Maximum value of powers of 2 mod n.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 4, 8, 8, 10, 8, 12, 8, 8, 8, 16, 16, 18, 16, 16, 20, 18, 16, 24, 24, 26, 16, 28, 16, 16, 16, 32, 32, 32, 32, 36, 36, 32, 32, 40, 32, 42, 40, 38, 36, 42, 32, 46, 48, 32, 48, 52, 52, 52, 32, 56, 56, 58, 32, 60, 32, 32, 32, 64, 64, 66, 64, 64

Offset: 1

Rémy Sigrist, Sep 21 2019

			For n = 10:
- the first powers of 2 mod 10 are:
    k   2^k mod 10
    --  ----------
     0           1
     1           2
     2           4
     3           8
     4           6
     5           2
- those values are eventually periodic, the maximum being 8,
- hence a(10) = 8.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, Colored scatterplot of the ordinal transform of the first 2^16 terms (colored pixels correspond to n's such that a(n) is a power of 2)

Cf. A000079, A062170, A047210, A327650.

f:= proc(n) local S,k,x,m;
  x:= 1; S:= {1}; m:= 1;
  for k from 1 do
    x:= 2*x mod n;
    if member(x,S) then return m fi;
    S:= S union {x};
    m:= max(m,x)
  od
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Feb 15 2023

a[n_] := PowerMod[2, Range[0, n-1], n] // Max;
Table[a[n], {n, 1, 1000}] (* Jean-François Alcover, May 14 2023 *)

a(n) = { my (p=1%n, mx=p); while (1, p=(2*p)%n; if (mx

a(2^k) = 2^(k-1) for any k > 0.
a(2^k+1) = 2^k for any k >= 0.
a(2^k-1) = 2^(k-1) for any k > 1.
If n = 2^j * r with r odd > 1 then a(n) = 2^j * a(r). - Robert Israel, Feb 15 2023

A327663 Maximum value of primorials mod n.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 6, 6, 6, 6, 8, 6, 9, 6, 6, 14, 15, 12, 18, 10, 9, 12, 15, 18, 20, 22, 24, 14, 23, 6, 30, 30, 30, 32, 30, 30, 30, 30, 30, 30, 35, 30, 38, 34, 30, 38, 44, 42, 42, 40, 42, 30, 51, 48, 45, 42, 48, 52, 58, 30, 60, 50, 42, 62, 35, 30, 62, 66, 48

Offset: 1

Rémy Sigrist, Sep 21 2019

nonn

			For n = 8:
- the first primorials mod 8 are:
    k  prime(k)#
    -  ---------
    0          1
    1          2
    2          6
    3          6
    4          2
- the prime numbers > 8 are of the form k + m*8, with m > 0 and k in {1, 3, 5, 7},
- starting from 2, and iteratively multiplying by a number in {1, 3, 5, 7} mod 8, we can only reach 2 or 6, and this value has already been reached before,
- hence a(8) = 6.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A327663

Cf. A002110, A062170.

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

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A221861 The least number k that maximizes k! mod n.

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A327649 Maximum value of powers of 2 mod n.

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A327663 Maximum value of primorials mod n.

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