A007843 -id:A007843 - cp's OEIS frontend

A210253 Number of distinct residues of all factorials mod 2^n.

1, 2, 3, 4, 5, 6, 8, 8, 9, 10, 11, 13, 14, 16, 16, 16, 17, 18, 19, 20, 22, 24, 24, 25, 26, 27, 29, 30, 32, 32, 32, 32, 33, 34, 35, 36, 37, 40, 40, 41, 42, 43, 45, 46, 48, 48, 48, 49, 50, 51, 52, 54, 56, 56, 57, 58, 59, 61, 62, 64, 64, 64, 64, 64, 65, 66, 67

Offset: 0

Vladimir Shevelev, Mar 19 2012

nonn

Theorem. For n>=1, a(n) = A007843(n) - A210255(n).

			Let n=2. We have modulo 4: 0!=1!==1, 2!==3!==2, for n>=4, n!==0. Thus the distinct residues are 0,1,2. Therefore, a(2) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000142, A007814, A210255.

a:= proc(n) local p, m, i, s;
      p:= 2^n;
      m:= 1;
      s:= {};
      for i to p while m<>0 do m:= m*i mod p; s:=s union {m} od;
      nops(s)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 20 2012

a[n_] := Module[{p = 2^n, m = 1, i, s = {}}, For[i = 1, i <= p && m != 0, i++, m = Mod[m i, p]; s = Union[s, {m}]]; Length[s]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz *)

A048842 Least positive integer k for which 13^n divides k!.

Original entry on oeis.org

1, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299, 312, 325, 338, 338, 351, 364, 377, 390, 403, 416, 429, 442, 455, 468, 481, 494, 507, 507, 520, 533, 546, 559, 572, 585, 598

Offset: 0

Charles T. Le (charlestle(AT)yahoo.com)

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

See A007843 for more information.

a(n) = {k = 1; ok = 0; until (ok, if (k! % 13^n == 0, ok=1, k++);); return (k);} \\ Michel Marcus, Jun 30 2013

A048843 Least positive integer k for which 17^n divides k!.

Original entry on oeis.org

1, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272, 289, 289, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459, 476, 493, 510, 527, 544, 561, 578, 578, 595, 612, 629, 646, 663, 680, 697, 714, 731, 748, 765, 782, 799

Offset: 0

Charles T. Le (charlestle(AT)yahoo.com)

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

See A007843 for more information.

A048844 Least positive integer k for which 19^n divides k!.

Original entry on oeis.org

1, 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 304, 323, 342, 361, 361, 380, 399, 418, 437, 456, 475, 494, 513, 532, 551, 570, 589, 608, 627, 646, 665, 684, 703, 722, 722, 741, 760, 779, 798, 817, 836, 855, 874, 893

Offset: 0

Charles T. Le (charlestle(AT)yahoo.com)

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

See A007843 for more information.

A048845 Least positive integer k for which 23^n divides k!.

Original entry on oeis.org

1, 23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483, 506, 529, 529, 552, 575, 598, 621, 644, 667, 690, 713, 736, 759, 782, 805, 828, 851, 874, 897, 920, 943, 966, 989, 1012, 1035, 1058

Offset: 0

Charles T. Le (charlestle(AT)yahoo.com)

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

See A007843 for more information.

A210337 Sum of distinct residues of all factorials mod 2^n.

Original entry on oeis.org

0, 1, 3, 9, 17, 49, 153, 281, 665, 1433, 3225, 7705, 17945, 47641, 64025, 129561, 293401, 752153, 1341977, 2914841, 6421017, 14547481, 33421849, 71170585, 138279449, 247331353, 645790233, 1182661145, 2558392857, 5779618329, 11685198361, 23496358425

Offset: 0

Vladimir Shevelev, Mar 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A007843, A210184, A210185.

a:= proc(n) local p, m, i, s;
      p:= 2^n;
      m:= 1;
      s:= {};
      for i to p while m<>0 do m:= m*i mod p; s:=s union {m} od;
      add(i, i=s)
    end:
seq (a(n), n=0..40);  # Alois P. Heinz, Mar 20 2012

a[n_] := Module[{k=0, s={}},  While[(r = Mod[k!, 2^n]) > 0, k++; AppendTo[s,r]]; Total@Union@s]; Array[a, 32, 0] (* Amiram Eldar, Dec 15 2018 *)

nbf(n) = my(k=1); while(k! % 2^n, k++); k; \\ A007843
a(n) = my(nb=nbf(n)); vecsum(Set(vector(nb, k, k! % 2^n))); \\ Michel Marcus, Dec 15 2018

More terms from Alois P. Heinz, Mar 20 2012

cp's OEIS Frontend

A210253 Number of distinct residues of all factorials mod 2^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A048842 Least positive integer k for which 13^n divides k!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A048843 Least positive integer k for which 17^n divides k!.

Views

Author

Keywords

Links

Crossrefs

A048844 Least positive integer k for which 19^n divides k!.

Views

Author

Keywords

Links

Crossrefs

A048845 Least positive integer k for which 23^n divides k!.

Views

Author

Keywords

Links

Crossrefs

A210337 Sum of distinct residues of all factorials mod 2^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions