A229743 -id:A229743 - cp's OEIS frontend

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

Leonid Broukhis, Mar 15 1996

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

			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.

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

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.

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

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

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

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

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

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

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

More terms from Benoit Cloitre, Jun 02 2002

Stolarsky reference from Matthew C. Russell, Oct 08 2013

A096303 Number of iterations of n -> n + (number of 1's in binary representation of n) needed for the trajectory of n to join the trajectory of A010062.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 0, 3, 2, 0, 1, 0, 2, 0, 1, 1, 0, 2, 0, 1, 6, 0, 2, 5, 0, 4, 1, 0, 3, 2, 0, 3, 2, 1, 1, 0, 5, 0, 2, 4, 0, 3, 1, 0, 2, 7, 0, 7, 1, 6, 1, 0, 5, 3, 0, 2, 4, 2, 1, 0, 3, 1, 6, 0, 0, 2, 0, 1, 5, 0, 2, 4, 0, 3, 1, 0, 2, 5, 0, 5, 1, 4, 1, 0, 3, 10, 0, 2, 2, 9, 1, 0, 1, 8, 1, 0, 8, 0, 7, 7, 0, 6, 6, 6, 0

Offset: 1

Jason Earls, Jun 25 2004

Conjecture: For any positive integer starting value n, iterations of n -> n + (number of 1's in binary representation of n) will eventually join A010062.

			a(6)=4 because the trajectory for 1 (sequence A010062) starts
1->2->3->5->7->10->12->14->17->19->22->25...
and the trajectory for 6 starts
6->8->9->11->14->17->19->22->25->28->31->36...
so the sequence beginning with 6 joins A010062 after 4 steps.

Cf. A010062.

For records see A229743, A229744.

a(n) = { my (o=1); for (k=0, oo, while (oRémy Sigrist, Apr 05 2020

cp's OEIS Frontend

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

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A096303 Number of iterations of n -> n + (number of 1's in binary representation of n) needed for the trajectory of n to join the trajectory of A010062.

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A229744 Values of records in A096303.

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