A004152 -id:A004152 - cp's OEIS frontend

A079584 Number of ones in the binary expansion of n!.

1, 1, 1, 2, 2, 4, 4, 6, 6, 6, 11, 7, 12, 12, 12, 18, 18, 22, 23, 17, 22, 25, 28, 31, 29, 30, 35, 38, 42, 40, 48, 42, 42, 46, 51, 56, 51, 58, 59, 64, 63, 66, 64, 71, 74, 70, 77, 81, 89, 87, 89, 90, 88, 94, 87, 99, 103, 98, 101, 109, 113, 103, 113, 120, 120, 109, 123, 121, 130, 121

Offset: 0

Jose R. Brox (tautocrona(AT)terra.es), Jan 26 2003

			a(5) = 4 because 5! = 120 and 120_10 = 1111000_2, with 4 ones.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Florian Luca, The number of non-zero digits of n!, Canad. Math. Bull. 45 (2002), pp. 115-118.
Carlo Sanna, On the sum of digits of the factorial, arXiv:1409.4912 [math.NT], 2014.
Carlo Sanna, On the sum of digits of the factorial, Journal of Number Theory 147 (2015), pp. 836-841.
Index entries for sequences related to factorial numbers

Cf. A000120 (binary weight), A000142 (factorial), A004152 (sum of decimal digits).

seq(convert(convert(n!,base,2),`+`), n=0..1000); # Robert Israel, Sep 18 2014

Table[DigitCount[n!,2,1],{n,70}] (* Harvey P. Dale, Jul 10 2012 *)

for(n=1,300,b=binary(n!); print1(sum(k=1,length(b),b[k])","))

a(n)=hammingweight(n!) \\ Charles R Greathouse IV, Mar 27 2013

import math
def a(n):
    return bin(math.factorial(n))[2:].count("1") # Indranil Ghosh, Dec 23 2016

a(n) << n log n. - Charles R Greathouse IV, Mar 27 2013

a(n) = A000120(A000142(n)). - Michel Marcus, Sep 18 2014

a(0)=1 prepended by Alois P. Heinz, Mar 07 2023

A066588 a(n) = sum of the digits of n^n.

Original entry on oeis.org

1, 1, 4, 9, 13, 11, 27, 25, 37, 45, 1, 41, 54, 58, 52, 99, 88, 98, 108, 127, 31, 117, 148, 146, 153, 151, 154, 189, 163, 167, 63, 184, 205, 207, 214, 260, 270, 271, 265, 306, 112, 308, 315, 313, 325, 306, 352, 374, 333, 355, 151, 414, 412, 350, 378, 442, 391, 450

Offset: 0

Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 07 2002

			a(7) = 25 because 7^7 = 823543 and 8 + 2 + 3 + 5 + 4 + 3 = 25.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000312, A007953, A004152.

[&+Intseq((n^n)): n in [0..80] ]; // Vincenzo Librandi, Jun 18 2015

a:= n-> add(i, i=convert(n^n, base, 10)):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 06 2023

Table[Plus@@IntegerDigits@(n^n), {n, 80}] (* Vincenzo Librandi, Jun 18 2015 *)

a(n) = sumdigits(n^n) \\ Michel Marcus, Jun 18 2015

More terms from Paolo P. Lava, May 15 2007

a(0)=1 inserted by Sean A. Irvine, Oct 06 2023

A120390 Sum of digits of double factorial numbers.

Original entry on oeis.org

1, 2, 3, 8, 6, 12, 6, 15, 18, 15, 18, 18, 18, 18, 18, 18, 36, 45, 45, 36, 45, 45, 36, 54, 63, 45, 72, 45, 72, 72, 90, 90, 90, 81, 108, 81, 108, 81, 126, 90, 108, 108, 144, 81, 144, 90, 135, 117, 144, 126, 153, 108, 180, 135, 180, 135, 171, 180, 180, 171, 198, 180, 198

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jun 30 2006

If n > 10, then 9 divides a(n). - Michel Lagneau, Dec 22 2011

			5!! = 5*3*1 = 15 --> 1+5 = 6

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A004152, A006882, A007953.

P:=proc(n) local i,j,k,t1,t2; for i from 1 by 1 to n do j:=i; k:=i-2; while k>0 do j:=j*k; k:=k-2; od; t1:=j; t2:=0; while t1 <> 0 do t2:= t2+(t1 mod 10); t1 := floor(t1/10); od; print(t2); od; end: P(100);

Table[Total[IntegerDigits[n!!]],{n,70}] (* Harvey P. Dale, May 19 2021 *)

a(n) = A007953(A006882(n)) - Eric Chen, Jun 13 2018

A131954 a(n) = sum of digits of (n! + a(n-1)), with a(1)=1.

Original entry on oeis.org

1, 3, 9, 6, 9, 18, 18, 18, 36, 36, 45, 36, 36, 54, 54, 72, 72, 63, 54, 63, 72, 81, 108, 90, 81, 90, 117, 99, 144, 126, 144, 117, 153, 153, 153, 180, 162, 117, 198, 207, 153, 198, 198, 234, 216, 225, 234, 243, 234, 225, 207, 288, 297, 279, 297, 351, 279, 306, 333, 297

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 31 2007

If n >= 5, then 9 divides a(n); see comment in A004152. - Bernard Schott, Jun 27 2019

			a(4) = Sum_digits(4!+9) = 6.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A004152, A055263, A131955.

P:=proc(n) local a,i,k,w; a:=0; for i from 1 by 1 to n do w:=0; k:=a+i!; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; a:=w; print(a); od; end: P(100);
# alternative:
sd:= n-> convert(convert(n,base,10),`+`):
A[1]:= 1:
for n from 2 to 100 do A[n]:= sd(n!+A[n-1]) od:
seq(A[i],i=1..100); # Robert Israel, Jun 26 2019

s={1};Do[AppendTo[s,DigitSum[n!+s[[-1]]]],{n,2,60}];s (* James C. McMahon, Mar 03 2025 *)

a(n) = Sum_digits(n!+a(n-1)).

Offset corrected by Robert Israel, Jun 26 2019

A066419 Numbers k such that k! is not divisible by the sum of the decimal digits of k!.

Original entry on oeis.org

432, 444, 453, 458, 474, 476, 485, 489, 498, 507, 509, 532, 539, 541, 548, 550, 552, 554, 555, 556, 560, 565, 567, 576, 593, 597, 603, 608, 609, 610, 611, 612, 613, 624, 630, 632, 634, 640, 645, 657, 663, 665, 683, 685, 686, 692, 698, 703, 706, 708, 714

Offset: 1

Matthew Conroy, Dec 25 2001

			The sum of the decimal digits of 5! is 1+2+0=3 and 3 divides 120, so 5 is not in the sequence.
The sum of the decimal digits of 432! is 3897 = (9)(433) and 3897 does not divide 432!, so 432 is in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Matthew Conroy)
Dimitri Zucker, Factorial Fact Frenzy (!), Combo Class Youtube video (2022).

Cf. A004152 (sum of digits of n!).

Select[Range[1000], !Divisible[Factorial[#],Total[IntegerDigits[Factorial[#]]]] &], (* Tanya Khovanova, Jun 13 2021 *)

isA066419(n) = (Mod(n!, sumdigits(n!)) != 0) \\ Jianing Song, Aug 26 2024

from math import factorial
def sd(n): return sum(map(int, str(n)))
def ok(f): return f%sd(f) != 0
print([n for n in range(1, 715) if ok(factorial(n))]) # Michael S. Branicky, Jun 13 2021

A086358 Digital root of n!.

Original entry on oeis.org

1, 1, 2, 6, 6, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Labos Elemer, Jul 21 2003

a(n) = 9 for n >= 6.

			n = 5, 5 != 120, iteration list = {120,3}, a(5) = 3.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000142, A086353-A086361, A007953, A010888, A038194, A029898, A004152.

sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]]; Table[FixedPoint[sud, w!], {w, 1, 87}]

a(n) = A010888(n!) = fixed-point of A007953(n!). It equals n! modulo(9); at r = 0 use 9.

G.f.: (1 + x^2 + 4*x^3 - 3*x^5 + 6*x^6)/(1 - x). - Stefano Spezia, Jan 26 2023

a(0) = 1 prepended by Alois P. Heinz, Dec 05 2018

A135204 Numbers n for which Sum_digits(n!) is a multiple of Sum_digits(n).

Original entry on oeis.org

1, 2, 3, 9, 10, 11, 12, 14, 16, 18, 20, 21, 22, 27, 28, 30, 33, 35, 36, 44, 45, 51, 54, 60, 61, 63, 72, 75, 81, 87, 90, 99, 100, 102, 105, 108, 111, 114, 117, 120, 126, 130, 135, 143, 144, 153, 158, 162, 165, 171, 180, 182, 185, 189, 190, 192, 200, 201, 202, 204, 206

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Nov 30 2007

I expect a(n) to be around kn log n for some constant k. - Charles R Greathouse IV, Apr 24 2013

			11 -> 11*10*9*8*7*6*5*4*3*2*1=39916800 -> (3+9+9+1+6+8+0+0)/(1+1)=18.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A004152, A120390, A108825, A129980, A131954, A131955, A135205, A135206.

P:=proc(n) local i,k,w,x; for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; x:=0; k:=i!; while k>0 do x:=x+k-(trunc(k/10)*10); k:=trunc(k/10); od; if trunc(x/w)=x/w then print(i); fi; od; end: P(1000);

Select[Range[100], Divisible[Total[IntegerDigits[#!, 10]], Total[IntegerDigits[#, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)

is(n)=sumdigits(n!)%sumdigits(n)==0 \\ Charles R Greathouse IV, Apr 24 2013

A135205 Numbers m for which Sum_digits(m!!) is a multiple of Sum_digits(m).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 10, 11, 12, 15, 18, 20, 21, 24, 25, 27, 30, 32, 33, 36, 42, 45, 46, 54, 55, 63, 72, 75, 81, 88, 90, 91, 93, 100, 101, 102, 105, 108, 111, 112, 117, 120, 121, 122, 123, 124, 126, 127, 135, 141, 144, 153, 154, 156, 162, 171, 176, 180, 182, 189, 198

Offset: 1

Paolo P. Lava, Nov 30 2007

			11 -> 11*9*7*5*3*1=10395 -> (1+0+3+9+5)/(1+1) = 9.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A004152, A120390, A108825, A129980, A131954, A131955, A135204, A135206.

P:=proc(n) local i,j,k,w,x; for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; x:=i; j:=i-2; while j >0 do x:=x*j; j:=j-2; od: k:=x; x:=0; while k>0 do x:=x+k-(trunc(k/10)*10); k:=trunc(k/10); od; if trunc(x/w)=x/w then print(i); fi; od; end: P(1000);

Select[Range[100], Divisible[Total[IntegerDigits[#!!, 10]], Total[IntegerDigits[#, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)

Offset 1 and b-file adapted by Paolo P. Lava, Jun 17 2024

A135206 Numbers m for which Sum_digits(m!) is a multiple of Sum_digits(m!!).

Original entry on oeis.org

1, 2, 3, 11, 19, 28, 48, 64, 158, 164, 190, 308, 324, 602, 782, 926, 1202, 1540, 1568, 1614, 2076, 2122, 2340, 2546, 2818, 2858, 2866, 3334, 3582, 3714, 4120, 4266, 4794, 5084, 5432, 5454, 5696, 6112, 6250, 6276, 6358, 6760, 7368, 8218, 8970, 9004, 9088

Offset: 1

Paolo P. Lava, Nov 30 2007

			11!=11*10*9*8*7*6*5*4*3*2*1=39916800 -> (3+9+9+1+6+8+0+0)=36,
11!!=11*9*7*5*3*1=10395 -> (1+0+3+9+5)=18,
36/18=2.

Michel Marcus, Table of n, a(n) for n = 1..90

Cf. A004152, A120390, A108825, A129980, A131954, A131955, A135204, A135205.

P:=proc(n) local i,j,k,w,x; for i from 1 by 1 to n do w:=0; k:=i!; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; x:=i; j:=i-2; while j >0 do x:=x*j; j:=j-2; od: k:=x; x:=0; while k>0 do x:=x+k-(trunc(k/10)*10); k:=trunc(k/10); od; if trunc(w/x)=w/x then print(i); fi; od; end: P(1000);

Select[Range[1000], Divisible[Total[IntegerDigits[#!, 10]], Total[IntegerDigits[#!!, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)

df(n) = prod(i=0, (n-1)\2, n - 2*i ); \\ A006882
isok(m) = !(sumdigits(m!) % sumdigits(df(m))); \\ Michel Marcus, Jun 18 2024

Changed offset to 1 by Paolo P. Lava, Jun 17 2024

A066235 Numbers k such that the sum of digits of k! is a square.

Original entry on oeis.org

0, 1, 6, 7, 8, 11, 24, 26, 33, 34, 35, 41, 47, 49, 59, 64, 76, 79, 116, 159, 167, 186, 253, 285, 314, 345, 376, 405, 413, 445, 459, 478, 480, 513, 520, 526, 676, 710, 769, 797, 833, 843, 852, 898, 937, 1004, 1032, 1043, 1098, 1192, 1291, 1365, 1478, 1491, 1496

Offset: 1

Jason Earls, Dec 19 2001

Harry J. Smith, Table of n, a(n) for n = 1..120

Cf. A004152.

Select[Range[0,1500],IntegerQ[Sqrt[Total[IntegerDigits[#!]]]]&] (* Harvey P. Dale, May 26 2020 *)

isok(m) = issquare(sumdigits(m!)) \\ Harry J. Smith, Feb 07 2010

cp's OEIS Frontend

A079584 Number of ones in the binary expansion of n!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A066588 a(n) = sum of the digits of n^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Extensions

A120390 Sum of digits of double factorial numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A131954 a(n) = sum of digits of (n! + a(n-1)), with a(1)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A066419 Numbers k such that k! is not divisible by the sum of the decimal digits of k!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A086358 Digital root of n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A135204 Numbers n for which Sum_digits(n!) is a multiple of Sum_digits(n).

Views

Author