A332063 a(1) = 1, a(n + 1) = a(n) + Sum_{k = 1..n} floor(log_2(a(k)) + 1): add total number of bits of the terms so far.

1, 2, 5, 11, 21, 36, 57, 84, 118, 159, 208, 265, 331, 406, 490, 583, 686, 799, 922, 1055, 1199, 1354, 1520, 1697, 1885, 2084, 2295, 2518, 2753, 3000, 3259, 3530, 3813, 4108, 4416, 4737, 5071, 5418, 5778, 6151, 6537, 6936, 7348, 7773, 8211, 8663, 9129, 9609, 10103, 10611

Offset

1,2

Comments

By number of bits we mean A070939 (length of base-2 representation), not the sum of nonzero bits or Hamming weight A000120.

One might consider starting this sequence with a(0) = 0, and defining (for the purpose of this sequence) log 0 = 0 as to get 1 for the number of bits of zero (although it is as well justified to consider 0 to have 0 bits). In that case one would get the sequence b(n) = a(n) + (n-1): (0, 1, 3, 7, 14, 25, 41, 63, 91, 126, 168, ...), similar to, but different from A004006.

Links

Table of n, a(n) for n=1..50.

Prog

(PARI) ({A332063_vec(N, a=1, s=-a)=vector(N, n, a+=s+=exponent(a)+1)})(50)

Crossrefs

Cf. A070939 (length of base-2 representation), A000120 (hammingweight).

Cf. A332064 for a variant where the number of bits is added or subtracted, depending on the parity of a(n).

Sequence in context: A328670 A294745 A352234 * A050407 A113032 A370722

Adjacent sequences: A332060 A332061 A332062 * A332064 A332065 A332066

Keyword

nonn

Author

M. F. Hasler, Feb 26 2020

Status

approved