A379537 Frugal numbers in base 2: numbers k such that A377369(k) < A070939(k).

1, 27, 32, 49, 64, 81, 121, 125, 128, 135, 147, 162, 169, 189, 192, 243, 250, 256, 289, 297, 320, 338, 343, 351, 361, 363, 375, 384, 405, 448, 486, 507, 512, 513, 529, 539, 567, 576, 578, 605, 621, 625, 637, 640, 648, 675, 686, 704, 722, 729, 750, 768, 783, 832

Offset: 1

Author

Paolo Xausa, Dec 25 2024

Keywords

• nonn
• base

Comments

A frugal number in base 2 is a number with more bits than the total number of bits of its prime factorization (including exponents > 1).

Following the definition by Pinch (1998), 1 is considered a frugal number.

Some authors call these numbers "economical numbers", as in A046759 which, according to the definition provided here, lists frugal numbers in base 10 (additionally, A046759 does not include 1).

Examples

			32 is a term because 32 = 2^5 = 10_2^101_2; the total number of bits of (10_2, 101_2) = 5 < the number of bits of 32 = 100000_2 (6).
135 is a term because 135 = 3^3*5 = 11_2^11_2*101_2; the total number of bits of (11_2, 11_2, 101_2) = 7 < the number of bits of 135 = 10000111_2 (8).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
• Wikipedia, Frugal number.

Crossrefs

Row n = 2 of A379538.

Cf. A046759, A070939, A377369, A379373.

Programs

Mathematica

A379537Q[k_] := Total[BitLength[Select[Flatten[FactorInteger[k]], # > 1 &]]] < BitLength[k];
Select[Range[1000], A379537Q]