A062762 Number of powerful numbers not exceeding 2^n.

1, 1, 2, 3, 5, 8, 11, 18, 26, 38, 55, 80, 116, 166, 240, 345, 497, 710, 1016, 1453, 2073, 2955, 4211, 5992, 8523, 12111, 17202, 24423, 34648, 49152, 69694, 98795, 140009, 198378, 281016, 398002, 563612, 797999, 1129737, 1599166, 2263457, 3203381

Offset

0,3

Comments

Number of terms x from A001694 for which x <= 2^n.

Links

Daniel Suteu, Table of n, a(n) for n = 0..90

Formula

a(n) = Sum_{k=0..n} A062761(k). - Daniel Suteu, Feb 18 2020

Example

Below 128, the 18 powerful numbers {1,4,8,9,16,25,...,100,108,121,125,128} can be found, so a(7)=18.

Mathematica

nn = 41; s = Union@ Flatten@ Table[a^2*b^3, {b, (2^nn)^(1/3)}, {a, Sqrt[(2^nn)/b^3]}]; Table[FirstPosition[s, 2^k][[1]], {k, 2, nn}] (* Michael De Vlieger, Oct 29 2023 *)

Prog

(PARI) a(n) = my(s=0, N=2^n); forsquarefree(k=1, sqrtnint(N, 3), s += sqrtint(N\k[1]^3)); s; \\ Daniel Suteu, Feb 18 2020

Crossrefs

Cf. A001694, A029837, A036380, A036386, A062761.

Sequence in context: A135908 A056891 A065462 * A004693 A272136 A254308

Adjacent sequences: A062759 A062760 A062761 * A062763 A062764 A062765

Keyword

nonn

Author

Labos Elemer, Jul 16 2001

Extensions

a(19)-a(41) from Donovan Johnson, Oct 01 2009

Status

approved