A078677 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = sum of these sums (including '1' and n itself).

1, 3, 6, 5, 8, 9, 13, 9, 12, 13, 17, 15, 19, 20, 20, 17, 20, 21, 25, 23, 27, 28, 28, 27, 31, 32, 32, 34, 34, 35, 39, 33, 36, 37, 41, 39, 43, 44, 44, 43, 47, 48, 48, 50, 50, 51, 55, 51, 55, 56, 56, 58, 58, 59, 63, 62, 62, 63, 67, 65, 69, 70, 72, 65, 68, 69, 73, 71, 75, 76, 76, 75

Offset: 1

Author

Frank Schwellinger (nummer_eins(AT)web.de), Dec 17 2002

Keywords

• base
• easy
• nonn

Examples

			a(13) = 19 because 13 = (1101) -> (1+1+0+1 = 11) -> (1+1 = 10) -> (1+0 = 1) = 1 and 1101+11+10+1 (binary) = 19 (decimal).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000120, A078627, A180094.

Row sums of A381962.

Programs

Mathematica

A078677[n_] := Total[NestWhileList[DigitSum[#, 2] &, n, # > 1 &]];
Array[A078677, 100] (* Paolo Xausa, Mar 11 2025 *)

Python

def a(n):
    s = n if n > 1 else 0
    while (n:=n.bit_count()) > 1:
        s += n
    return s + 1
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Mar 12 2025

Formula

a(1) = 1; for n > 1, a(n) = n + a(A000120(n)).

a(n) = Sum_{k = 0..A180094(n)} A381962(n,k). - Paolo Xausa, Mar 12 2025