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

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

Author

Labos Elemer, May 27 2002

Keywords

• base
• easy
• nonn

Links

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

Crossrefs

Cf. A000142, A000788.

Partial sums of A079584 starting at index 1.

Programs

Mathematica

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 *)

PARI

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

Python

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

Formula

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