A188064 Partial sums of wt(n)! where wt(n) is the Hamming weight of n (A000120).

1, 2, 3, 5, 6, 8, 10, 16, 17, 19, 21, 27, 29, 35, 41, 65, 66, 68, 70, 76, 78, 84, 90, 114, 116, 122, 128, 152, 158, 182, 206, 326, 327, 329, 331, 337, 339, 345, 351, 375, 377, 383, 389, 413, 419, 443, 467, 587, 589, 595, 601, 625, 631, 655, 679, 799, 805, 829, 853, 973, 997, 1117

Offset: 0

Author

Joerg Arndt, Mar 20 2011

Keywords

• nonn

Comments

Partial sums of A093659, partial sums of the factorials of A000120.

A000522 is a subsequence: A000522(n)=a(2^n-1).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A093659, A000120, A000522.

Programs

Mathematica

FoldList[Plus, 0!, Table[(Plus @@ IntegerDigits[n, 2])!, {n, 1, 70}]] (* From Olivier Gérard, Mar 23 2011 *)
Accumulate[DigitCount[Range[0,70],2,1]!] (* Harvey P. Dale, Jun 26 2013 *)

PARI

bitcount(x)=
{ /* Return Hamming weight of x */
    local(p);  p = 0;
    while ( x, p+=bitand(x, 1); x>>=1; );
    return( p );
}
N=65; /* that many terms */
f=vector(N,n,bitcount(n-1)!); /* factorials of Hamming weights */
s=vector(N); s[1]=f[1]; /* for cumulative sums */
for (n=2,N,s[n]=s[n-1]+f[n]); /* sum up */
s /* show terms */ /* Joerg Arndt, Mar 20 2011 */

Formula

a(n)=sum(k=0,n,wt(k)!) where wt(k) is the Hamming weight of k.