A079584 -id:A079584 - cp's OEIS frontend

A004152 Sum of digits of n!.

1, 1, 2, 6, 6, 3, 9, 9, 9, 27, 27, 36, 27, 27, 45, 45, 63, 63, 54, 45, 54, 63, 72, 99, 81, 72, 81, 108, 90, 126, 117, 135, 108, 144, 144, 144, 171, 153, 108, 189, 189, 144, 189, 180, 216, 207, 216, 225, 234, 225, 216, 198, 279, 279, 261, 279, 333, 270, 288

Offset: 0

N. J. A. Sloane

If n > 5, then 9 divides a(n). - Enrique Pérez Herrero, Mar 01 2009

			a(5) = 3 because 5! = 120 and 1 + 2 + 0 = 3.
a(6) = 9 because 6! = 720 and 7 + 2 + 0 = 9.

Maciej Ireneusz Wilczynski, 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, Journal of Number Theory 147 (February 2015), pp. 836-841. arXiv:1409.4912 [math.NT].
Carlo Sanna, On the sum of digits of the factorial, Journal of Number Theory 147 (February 2015), pp. 836-841.

Cf. A000142 (factorial), A007953 (sum of digits), A079584 (same in base 2), A086358 (digital root of n!).

Cf. A066419 (k such that a(k) does not divide k!).

[&+Intseq(Factorial(n)): n in [0..70]]; // Vincenzo Librandi, Jan 30 2015

seq(convert(convert(n!,base,10),`+`),n=0..100); # Robert Israel, Nov 13 2014

Table[ Plus @@ IntegerDigits[n!], {n, 0, 100}] (* Enrique Pérez Herrero, Mar 01 2009 *)

a(n)=my(v=eval(Vec(Str(n!))));sum(i=1,#v,v[i]) \\ Charles R Greathouse IV, Dec 27 2011

a(n) = sumdigits(n!); \\ Michel Marcus, Sep 18 2014

Luca shows that a(n) >> log n. In particular, a(n) > log_10 n - log_10 log_10 n. - Charles R Greathouse IV, Dec 27 2011
a(n) < floor(log_10(n)*9/2). - Carmine Suriano, Feb 20 2013
a(n) = A007953(A000142(n)). - Michel Marcus, Sep 18 2014
a(n) < 9*(A034886(n) - A027868(n)). - Enrique Pérez Herrero, Nov 16 2014
Sanna improved Luca's result to a(n) >> log n log log log n. - Charles R Greathouse IV, Jan 30 2015
a(n) = 9*A202708(n), n>=6. - R. J. Mathar, Jul 30 2021

A095375 Total number of 1's in the binary expansions of the first n primes: summatory A014499.

Original entry on oeis.org

1, 3, 5, 8, 11, 14, 16, 19, 23, 27, 32, 35, 38, 42, 47, 51, 56, 61, 64, 68, 71, 76, 80, 84, 87, 91, 96, 101, 106, 110, 117, 120, 123, 127, 131, 136, 141, 145, 150, 155, 160, 165, 172, 175, 179, 184, 189, 196, 201, 206, 211, 218, 223, 230, 232, 236, 240, 245, 249, 253

Offset: 1

Labos Elemer, Jun 07 2004

			n=4: first 4 primes={10,11,101,111}, with a(4)=8 digits 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A000788, A079584, A014499.

read("transforms") :
A095375 := proc(n)
    local a;
    a := 0 ;
    for i from 1 to n do
        a := a+wt(ithprime(i)) ;
    end do:
end proc: # R. J. Mathar, Jul 13 2012
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)
      +add(i, i=Bits[Split](ithprime(n))))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 26 2021

lib[x_] :=Count[IntegerDigits[x, 2], 1] {s=0, ta=Table[0, {256}]}; Do[s=s+lib[Prime[n]]; ta[[n]]=s, {n, 1, 256}] ta

a(n)=my(s);forprime(p=2,prime(n),s+=hammingweight(p));s \\ Charles R Greathouse IV, Mar 29 2013

from sympy import primerange, prime
def A095375(n): return sum(p.bit_count() for p in primerange(prime(n)+1)) # Chai Wah Wu, Nov 12 2024

A078565 Number of zeros in the binary expansion of n!.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 10, 13, 11, 19, 17, 21, 25, 23, 27, 27, 30, 40, 40, 41, 42, 44, 51, 54, 54, 56, 56, 63, 60, 71, 76, 77, 77, 77, 88, 86, 90, 90, 97, 99, 106, 105, 107, 117, 115, 117, 114, 122, 126, 130, 138, 138, 151, 144, 146, 157, 160, 158, 160, 176

Offset: 0

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

			a(4) = 3 because 4! = 24_10 = 11000_2 and it has 3 zero digits.

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

Cf. A072831, A079584.

Table[Count[IntegerDigits[n!, 2], 0], {n, 0, 100}] (* T. D. Noe, Apr 10 2012 *)
DigitCount[Range[0,70]!,2,0] (* Harvey P. Dale, Mar 11 2019 *)

for(n=1,300,b=binary(n!); print1(sum(k=1,length(b),if(b[k],0,1))","))

import math
def a(n):
    return bin(math.factorial(n))[2:].count("0")
# Indranil Ghosh, Dec 23 2016

a(n) = A072831(n) - A079584(n). - Michel Marcus, Dec 23 2016

A379151 The binary weights of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 2, 6, 6, 9, 6, 8, 8, 12, 9, 16, 13, 17, 12, 17, 13, 18, 15, 25, 20, 17, 20, 24, 28, 25, 26, 25, 25, 32, 27, 34, 29, 32, 33, 29, 35, 29, 31, 36, 35, 44, 44, 49, 40, 46, 48, 44, 38, 50, 43, 44, 46, 47, 55, 50, 52, 58, 59, 60, 65, 68, 56, 62, 68

Offset: 0

Amiram Eldar, Dec 16 2024

			a(10) = 6 because Catalan(10) = 16796 = 100000110011100_2, which has 6 one bits. - _Vincenzo Librandi_, Feb 05 2025

Amiram Eldar, Table of n, a(n) for n = 0..10000
Amiram Eldar, Plot of (1/n^2) * Sum_{k=1..n} a(k) for n = 2^(8..23).
Florian Luca and Igor E. Shparlinski, On the g-ary expansions of middle binomial coefficients and Catalan numbers, The Rocky Mountain Journal of Mathematics, Vol. 41, No. 4 (2011), pp. 1291-1301.

Cf. A000102, A000108.

Similar sequences: A011373, A079584, A082481, A379152, A379153.

[&+Intseq(Catalan(n), 2): n in [0..100]]; // Vincenzo Librandi, Feb 05 2025

a[n_] := DigitCount[CatalanNumber[n], 2, 1]; Array[a, 100, 0]

a(n) = hammingweight(binomial(2*n, n)/(n+1));

a(n) = A000120(A000108(n)).
Two formulas from Luca and Shparlinski (2011):
a(n) >= 3 for all but finitely many positive integers n.
a(n) >> eps(n) * sqrt(log(n)), for all n <= X with at most o(X) exceptions as X -> oo, where eps(n) is a function tending to zero as n -> oo.
Conjecture: Sum_{k=1..n} a(k) ~ n^2 / 2 (see the plot in the Links section).

A379153 The binary weights of the Apéry numbers (A005259).

Original entry on oeis.org

1, 2, 3, 6, 6, 14, 15, 15, 20, 19, 23, 23, 27, 34, 35, 44, 40, 36, 40, 44, 41, 48, 52, 62, 64, 66, 57, 66, 72, 79, 71, 75, 77, 78, 79, 78, 88, 86, 92, 100, 103, 103, 92, 116, 96, 116, 117, 113, 129, 117, 123, 128, 123, 126, 130, 133, 129, 142, 147, 134, 135, 148

Offset: 0

Amiram Eldar, Dec 17 2024

Amiram Eldar, Table of n, a(n) for n = 0..10000
Arnold Knopfmacher and Florian Luca, Digit sums of binomial sums, Journal of Number Theory, Vol. 132, No. 2 (2012), pp. 324-331.
Florian Luca and Igor E. Shparlinski, On the g-ary expansions of Apéry, Motzkin, Schröder and other combinatorial numbers, Annals of Combinatorics, Vol. 14 (2010), pp. 507-524.

Cf. A000120, A005259.

Similar sequences: A011373, A079584, A082481, A379151, A379152.

a[n_] := DigitCount[Sum[(Binomial[n, k] * Binomial[n+k, k])^2, {k, 0, n}], 2, 1]; Array[a, 100, 0]

a(n) = hammingweight(sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2));

a(n) = A000120(A005259(n)).
a(n) > c * (log(n)/log(log(n)))^(1/4) holds on a set of n of asymptotic density 1, where c > 0 is a constant (Luca and Shparlinski, 2010).
a(n) > c * log(n)/log(log(n)) holds on a set of n of asymptotic density 1, where c > 0 is a constant (Knopfmacher and Luca, 2012).
Conjecture: Limit_{m->oo} (1/m^2) * Sum_{k=1..m} a(k) = log(sqrt(2) + 1)/log(2) = 1.2715533... (Knopfmacher and Luca, 2012).

A214561 Number of 1's in binary expansion of n^n.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 6, 11, 1, 14, 11, 20, 10, 28, 23, 33, 1, 31, 27, 41, 26, 49, 36, 59, 16, 58, 41, 68, 37, 62, 51, 83, 1, 79, 61, 88, 58, 97, 85, 98, 53, 115, 96, 116, 63, 123, 96, 128, 41, 138, 105, 144, 90, 163, 128, 164, 81, 172, 148, 181, 124, 167, 134, 201, 1

Offset: 0

Alex Ratushnyak, Jul 21 2012

nonn

a(n) + A214562(n) = 1+floor(log_2(n^n)) = 1, 1, 3, 5, 9, 12, 16, 20, 25, 29, 34, 39, 44, 49... is the number of binary digits in n^n. - R. J. Mathar, Jul 22 2012

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A000120, A159918, A079584.

a:= proc(n) option remember; local m, r;
      m, r:= n^n, 0;
      while m>0 do r:= r +irem(m, 2, 'm') od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 21 2012

Table[Count[IntegerDigits[n^n,2],1],{n,1,64}] (* Geoffrey Critzer, Sep 30 2013 *)
Join[{1},Table[DigitCount[n^n,2,1],{n,100}]] (* Harvey P. Dale, Oct 13 2022 *)

vector(66, n, b=binary((n-1)^(n-1)); sum(j=1, #b, b[j])) /* Joerg Arndt, Jul 21 2012 */

for n in range(300):
    c = 0
    b = n**n
    while b>0:
        c += b&1
        b//=2
    print(c, end=',')

def a(n): return bin(n**n)[2:].count('1')
print([a(n) for n in range(65)]) # Michael S. Branicky, May 22 2021

a(n) = A000120(A000312(n)).

a(2^k)=1.

A354300 Numbers k such that k! and (k+1)! have the same binary weight (A000120).

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 12, 13, 15, 31, 63, 88, 127, 129, 131, 244, 255, 262, 263, 288, 300, 344, 511, 793, 914, 1012, 1023, 1045, 1116, 1196, 1538, 1549, 1565, 1652, 1817, 1931, 1989, 2047, 2067, 2096, 2459, 2548, 2862, 2918, 2961, 3372, 3478, 3540, 3588, 3673, 3707

Offset: 1

Amiram Eldar, May 23 2022

Numbers k such that A079584(k) = A079584(k+1).
The corresponding values of A079584(k) are 1, 1, 2, 4, 6, 6, 12, 12, 18, 42, ...
This sequence is infinite as it contains A000225. - Rémy Sigrist, May 23 2022

			1 is a term since A079584(1) = A079584(2) = 1.
3 is a term since A079584(3) = A079584(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..1000

A354301 is a subsequence.

Cf. A000120, A000142, A000225, A079584, A353986.

s[n_] := s[n] = DigitCount[n!, 2, 1]; Select[Range[0, 4000], s[#] == s[# + 1] &]

isok(k) = hammingweight(k!) == hammingweight((k+1)!); \\ Michel Marcus, May 23 2022

from itertools import count, islice
def wt(n): return bin(n).count("1")
def agen(): # generator of terms
    n, fn, fnplus, wtn, wtnplus = 0, 1, 1, 1, 1
    for n in count(0):
        if wtn == wtnplus: yield n
        fn, fnplus = fnplus, fnplus*(n+2)
        wtn, wtnplus = wtnplus, wt(fnplus)
print(list(islice(agen(), len(data)))) # Michael S. Branicky, May 23 2022

A379152 The binary weights of the odd Catalan numbers.

Original entry on oeis.org

1, 1, 2, 6, 16, 25, 60, 127, 244, 494, 1010, 2015, 4076, 8086, 16281, 32818, 65518, 131059, 262348, 524448, 1047643, 2097675, 4194133, 8386693, 16776916, 33554390, 67114125, 134214652, 268452748

Offset: 0

Amiram Eldar, Dec 17 2024

Florian Luca and Paul Thomas Young, On the binary expansion of the odd Catalan numbers, Proceedings of the XIVth International Conference on Fibonacci Numbers, Morelia, Mexico, 2010, pp. 185-190; ResearchGate link.

Cf. A000108, A000120, A038003.

Similar sequences: A011373, A079584, A082481, A379151, A379153.

a[n_] := DigitCount[CatalanNumber[2^n-1], 2, 1]; Array[a, 23, 0]

a(n) = my(m = -1 + 1 << n); hammingweight(binomial(2*m, m)/(m+1));

from itertools import count, islice
def A379152_gen(): # generator of terms
    yield from [1,1]
    c, s = 1, 3
    for n in count(2):
        c = (c*((n<<2)-2))//(n+1)
        if n == s:
            yield c.bit_count()
            s = (s<<1)|1
A379152_list = list(islice(A379152_gen(),10)) # Chai Wah Wu, Dec 17 2024

a(n) = A000120(A038003(n)) = A000120(A000108(2^n-1)).

a(n) = A379151(2^n-1).

A071425 Total number of 1's in binary representation of all factorials from 1 to n.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 20, 26, 32, 43, 50, 62, 74, 86, 104, 122, 144, 167, 184, 206, 231, 259, 290, 319, 349, 384, 422, 464, 504, 552, 594, 636, 682, 733, 789, 840, 898, 957, 1021, 1084, 1150, 1214, 1285, 1359, 1429, 1506, 1587, 1676, 1763, 1852, 1942, 2030, 2124

Offset: 1

Labos Elemer, May 27 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000142, A000788.

Partial sums of A079584 starting at index 1.

s=0; Do[s=s+Apply[Plus, IntegerDigits[n!, 2]]; Print[s], {n, 1, 128}]
Accumulate[DigitCount[Range[60]!,2,1]] (* Harvey P. Dale, Apr 18 2014 *)

list(lim) = {my(s = 0); for(k = 1, lim, s += hammingweight(k!); print1(s, ", "));} \\ Amiram Eldar, Mar 18 2025

def A071425(n):
    c, a = 0, 1
    for i in range(1,n+1):
        c += (a:=a*i).bit_count()
    return c # Chai Wah Wu, Nov 12 2024

a(n) = Sum_{i=1..n} A079584(i).

A348651 Number of ones in the binary expansion of (n!)!.

Original entry on oeis.org

1, 1, 1, 4, 29, 293, 2566, 24844, 259437, 2908263, 35102629, 455204360, 6321171774

Offset: 0

Alois P. Heinz, Oct 27 2021

			a(3) = 4 because (3!)! = 6! = 720 = 1011010000_2 which has 4 ones.

Index entries for sequences related to factorial numbers

Cf. A000120, A000142, A000197, A079584, A152168, A301861.

a:= n-> add(i, i=Bits[Split](n!!)):
seq(a(n), n=0..10);

a[n_] := DigitCount[(n!)!, 2, 1]; Array[a, 10, 0] (* Amiram Eldar, Oct 29 2021 *)

a(n) = hammingweight((n!)!); \\ Michel Marcus, Oct 29 2021

from gmpy2 import fac, popcount
def A348651(n): return popcount(fac(fac(n))) # Chai Wah Wu, Oct 28 2021

a(n) = A000120(A000197(n)).

a(11)-a(12) from Chai Wah Wu, Oct 28 2021

cp's OEIS Frontend

A004152 Sum of digits of n!.

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A095375 Total number of 1's in the binary expansions of the first n primes: summatory A014499.

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A078565 Number of zeros in the binary expansion of n!.

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A379151 The binary weights of the Catalan numbers (A000108).

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A379153 The binary weights of the Apéry numbers (A005259).

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A214561 Number of 1's in binary expansion of n^n.

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A354300 Numbers k such that k! and (k+1)! have the same binary weight (A000120).

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