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

%I A379537 #18 Jan 01 2025 18:13:29
%S A379537 1,27,32,49,64,81,121,125,128,135,147,162,169,189,192,243,250,256,289,
%T A379537 297,320,338,343,351,361,363,375,384,405,448,486,507,512,513,529,539,
%U A379537 567,576,578,605,621,625,637,640,648,675,686,704,722,729,750,768,783,832
%N A379537 Frugal numbers in base 2: numbers k such that A377369(k) < A070939(k).
%C A379537 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).
%C A379537 Following the definition by Pinch (1998), 1 is considered a frugal number.
%C A379537 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).
%H A379537 Paolo Xausa, <a href="/A379537/b379537.txt">Table of n, a(n) for n = 1..10000</a>
%H A379537 Richard G. E. Pinch, <a href="https://doi.org/10.48550/arXiv.math/9802046">Economical numbers</a>, arXiv:math/9802046 [math.NT], 1998.
%H A379537 Wikipedia, <a href="https://en.wikipedia.org/wiki/Frugal_number">Frugal number</a>.
%e A379537 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).
%e A379537 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).
%t A379537 A379537Q[k_] := Total[BitLength[Select[Flatten[FactorInteger[k]], # > 1 &]]] < BitLength[k];
%t A379537 Select[Range[1000], A379537Q]
%Y A379537 Row n = 2 of A379538.
%Y A379537 Cf. A046759, A070939, A377369, A379373.
%K A379537 nonn,base
%O A379537 1,2
%A A379537 _Paolo Xausa_, Dec 25 2024