A322149 In the binary representation of n, replace each run of k 0's (or 1's) with k^2 0's (or 1's).

0, 1, 2, 15, 16, 5, 30, 511, 512, 33, 10, 47, 240, 61, 1022, 65535, 65536, 1025, 66, 271, 80, 21, 94, 1535, 7680, 481, 122, 495, 8176, 2045, 131070, 33554431, 33554432, 131073, 2050, 8207, 528, 133, 542, 8703, 2560, 161, 42, 175, 752, 189, 3070, 196607, 983040

Offset: 0

Author

Rémy Sigrist, Nov 28 2018

Keywords

• nonn
• base

Comments

This sequence has similarities with A001196: here we square the length of each run of consecutive equal bits, there we double it.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..16383
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000975, A001196, A005811, A322403.

Programs

Mathematica

squareList[v_] := Flatten[ConstantArray[v, {Length[v]}]]; a[n_] := FromDigits[ Flatten[squareList /@ Split[IntegerDigits[n, 2]]], 2]; Array[a, 60, 0] (* Amiram Eldar, Dec 07 2018*)

PARI

a(n) = if (n==0, 0, my (b=n%2, k=valuation(n+b,2)); (a(n\2^k) + b) * 2^(k^2) - b)

Formula

a(n) = A322403(n, n).

a(n) >= n with equality iff n belongs to A000975.

a(2^n) = 2^(n^2) for any n >= 0.

a(2^n - 1) = 2^(n^2) - 1 for any n >= 0.

A005811(a(n)) = A005811(n).