A008935 If 2n = Sum 2^e(k) then a(n) = Sum e(k)^2.

1, 4, 5, 9, 10, 13, 14, 16, 17, 20, 21, 25, 26, 29, 30, 25, 26, 29, 30, 34, 35, 38, 39, 41, 42, 45, 46, 50, 51, 54, 55, 36, 37, 40, 41, 45, 46, 49, 50, 52, 53, 56, 57, 61, 62, 65, 66, 61, 62, 65, 66, 70, 71, 74, 75, 77, 78, 81, 82, 86, 87, 90, 91, 49, 50, 53, 54, 58, 59, 62

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Examples

			To get a(5), we write 10 = 2 + 8 = 2^1 + 2^3 so a(5) = 1^2 + 3^2 = 10.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1023

Crossrefs

Gives A003995 if sorted and duplicates removed.

Programs

C

#include 
#include 
#define USAGE "Usage: 'A008935 num'\n where num is the index of the desired ending value in the sequence.\n"
#define MAX 1023
#define SHIFT_MAX 9
int main(int argc, char *argv[]) { unsigned short mask, i, j, end; unsigned long sum; if (argc < 2) { fprintf(stderr, USAGE); return EXIT_FAILURE; } end = atoi(argv[1]); end = (end >= MAX) ? MAX : end;
fprintf(stdout, "Values: "); for (i = 1; i <= end; i++) { sum = 0; mask = 1; for (j = 0; j < SHIFT_MAX; j++, mask *= 2) if (i & mask) sum += (j+1) * (j+1); fprintf(stdout, "%ld", sum); if (i < end) fprintf(stdout, ","); } fprintf(stdout, "\n"); return EXIT_SUCCESS; }

Haskell

a008935 = f 1 where
   f k x | x == 0    = 0
         | r == 0    = f (k+1) x'
         | otherwise = k^2 + f (k+1) x' where (x',r) = divMod x 2
-- Reinhard Zumkeller, Jul 05 2011

Maple

a:= n-> (l-> add(l[i]*i^2, i=1..nops(l)))(convert(n, base, 2)):
seq(a(n), n=1..80);  # Alois P. Heinz, Nov 20 2019

Mathematica

a[n_] := Total[Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]]^2]; Table[a[n],{n,1,70}] (* Jean-François Alcover  Mar 21 2011 *)

Python

a = lambda n: sum(((k+1)**2) * ((n >> k) & 1) for k in range(0, n.bit_length()))
print([a(n) for n in range(1,68)]) # Darío Clavijo, Dec 27 2024

Formula

G.f.: 1/(1-x) * Sum_{k>=0} (k+1)^2*x^2^k/(1+x^2^k). - Ralf Stephan, Jun 23 2003

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Mar 22 2000