OFFSET
0,2
COMMENTS
Sequence A230297 (and A157845 without initial term) converted from binary to decimal, cf. formula. - M. F. Hasler, Nov 18 2019
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)
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
EXAMPLE
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.
MATHEMATICA
NestList[# + DigitCount[#, 2, 1] &, 1, 60] (* Alonso del Arte, Oct 26 2012 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn,base,easy,nice
AUTHOR
Leonid Broukhis, Mar 15 1996
EXTENSIONS
More terms from Benoit Cloitre, Jun 02 2002
Stolarsky reference from Matthew C. Russell, Oct 08 2013
STATUS
approved