A309759 Numbers that are sums of consecutive powers of 4.

1, 4, 5, 16, 20, 21, 64, 80, 84, 85, 256, 320, 336, 340, 341, 1024, 1280, 1344, 1360, 1364, 1365, 4096, 5120, 5376, 5440, 5456, 5460, 5461, 16384, 20480, 21504, 21760, 21824, 21840, 21844, 21845, 65536, 81920, 86016, 87040, 87296, 87360, 87376, 87380

Offset

1,2

Comments

Numbers of the form (4^i - 4^j)/3 with i > j.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

336 = 4^2 + 4^3 + 4^4, so 336 is in the sequence.
+------+--------+
| a(n) | base 4*|
+------+--------+
|   1  |     1  |
|   4  |    10  |
|   5  |    11  |
|  16  |   100  |
|  20  |   110  |
|  21  |   111  |
|  64  |  1000  |
|  80  |  1100  |
|  84  |  1110  |
|  85  |  1111  |
+------+--------+
* - a(n) written in base 4.

Mathematica

Module[{nn=10, k}, k=4^Range[0, nn]; Table[Accumulate[Reverse[Take[k, n]]], {n, nn}]]//Flatten (* Harvey P. Dale, May 29 2021 *)

Prog

(Python)
from math import isqrt
def A309759(n): return ((1<<((m:=isqrt(n<<3)+1>>1)<<1))-(1<<m*(m+1)-(n<<1)&-2))//3 # Chai Wah Wu, Apr 04 2025

Crossrefs

Cf. A000302, A000695, A023758, A038470, A309758, A309761.

Sequence in context: A081345 A137527 A024854 * A025617 A227855 A078581

Adjacent sequences: A309756 A309757 A309758 * A309760 A309761 A309762

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 15 2019

Status

approved