A010062 a(0)=1; thereafter a(n+1) = a(n) + number of 1's in binary representation of a(n).

1, 2, 3, 5, 7, 10, 12, 14, 17, 19, 22, 25, 28, 31, 36, 38, 41, 44, 47, 52, 55, 60, 64, 65, 67, 70, 73, 76, 79, 84, 87, 92, 96, 98, 101, 105, 109, 114, 118, 123, 129, 131, 134, 137, 140, 143, 148, 151, 156, 160, 162, 165, 169, 173, 178, 182, 187, 193, 196, 199, 204

Offset: 0

Author

Leonid Broukhis, Mar 15 1996

Keywords

• nonn
• base
• easy
• nice

Comments

Sequence A230297 (and A157845 without initial term) converted from binary to decimal, cf. formula. - M. F. Hasler, Nov 18 2019

Examples

			a(7) = 14 because a(6) = 12, which is 1100 in binary (having 2 on bits), and 12 + 2 = 14.
a(8) = 17 because a(7) = 14, which is 1110 in binary (having 3 on bits), and 14 + 3 = 17.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Raoul Nakhmanson-Kulish, Graph of a(n)/(n*log_2(n)/2), showing self-similar fractal structure.
• Raoul Nakhmanson-Kulish, Graph of f(n), where f(n) = (a(n)-n*log_2(n)/2)/(n*sqrt(log_2(n)*log_2 log_2(n))) (see Stolarsky's estimate below).
• Kenneth B. Stolarsky, The sum of a digitaddition series, Proc. Amer. Math. Soc. 59 (1976), no. 1, 1--5. MR0409340 (53 #13099)
• Index entries for Colombian or self numbers and related sequences

Crossrefs

First row of A228083.

For the base-10 analog see A004207.

Cf. A000120, A010061, A092391, A229167, A096303, A229743, A229744, A230297 (this sequence written in binary), A230298 (read mod 2).

See A230088 for partial sums.

Equals A028897 o A230297 = A028897 o A157845 (up to offset); see also A007088.

Programs

Haskell

a010062 n = a010062_list !! n
a010062_list = iterate a092391 1  -- Reinhard Zumkeller, May 13 2012

Magma

[n le 1 select 1 else Self(n-1)+&+Intseq(Self(n-1),2): n in [1..61]]; // Bruno Berselli, Oct 27 2012

Mathematica

NestList[# + DigitCount[#, 2, 1] &, 1, 60] (* Alonso del Arte, Oct 26 2012 *)

PARI

print1(s=1);for(n=2,30,print1(", ", s+=hammingweight(s))) \\ Charles R Greathouse IV, Oct 27 2012

PARI

A010062=List(1); A010062(n)={for(n=#A010062,n, listput(A010062, A092391(A010062[n])));A010062[n+1]} \\ A092391(n)=n+hammingweight(n) \\ M. F. Hasler, Nov 18 2019

Python

from itertools import islice
def agen():
    an = 1
    while True: yield an; an += an.bit_count()
print(list(islice(agen(), 61))) # Michael S. Branicky, Jul 31 2022

Formula

a(n) = (n/2)*log n + O(n*sqrt(log n * loglog n)), where log means log_2. In particular, a(n) ~ (n/2)*log n. [Stolarsky]

a(n + 1) = A092391(a(n)) = a(n) + A000120(a(n)). - Reinhard Zumkeller, May 27 2012, May 08 2004; corrected thanks to a notice by Lambert Herrgesell

a(n) = A028897(A230297(n)) = A028897(A157845(n+1)). - M. F. Hasler, Nov 18 2019

Extensions

More terms from Benoit Cloitre, Jun 02 2002

Stolarsky reference from Matthew C. Russell, Oct 08 2013