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

1,2

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).

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.

Example

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).

Mathematica

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

Crossrefs

Row n = 2 of A379538.

Cf. A046759, A070939, A377369, A379373.

Sequence in context: A324248 A377478 A275188 * A198147 A144862 A117934

Adjacent sequences: A379534 A379535 A379536 * A379538 A379539 A379540

Keyword

nonn,base

Author

Paolo Xausa, Dec 25 2024

Status

approved