This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379538 #24 Jan 01 2025 09:49:42
%S A379538 1,1,27,1,32,32,1,27,49,49,1,27,64,64,64,1,81,81,81,81,81,1,64,125,
%T A379538 125,121,98,121,1,64,81,243,128,125,121,125,1,81,81,125,250,162,128,
%U A379538 125,128,1,125,125,125,243,256,169,169,128,135,1,125,128,128,128,343,289,243,243,169,147
%N A379538 Square array read by ascending antidiagonals: T(n,k) is the k-th frugal number in base n.
%C A379538 A frugal number in base n is a number with more digits (in its base n representation) than the total number of digits (in base n representation) of its prime factorization (including exponents > 1).
%C A379538 Following the definition by Pinch (1998), 1 is considered a frugal number.
%C A379538 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 A379538 Richard G. E. Pinch, <a href="https://doi.org/10.48550/arXiv.math/9802046">Economical numbers</a>, arXiv:math/9802046 [math.NT], 1998.
%H A379538 Giovanni Resta, <a href="https://www.numbersaplenty.com/set/frugal_number/">Frugal numbers</a>, Numbers Aplenty, 2013.
%H A379538 Wikipedia, <a href="https://en.wikipedia.org/wiki/Frugal_number">Frugal number</a>.
%e A379538 Array begins:
%e A379538 n\k| 1 2 3 4 5 6 7 8 9 10 ...
%e A379538 ---------------------------------------------------------
%e A379538 2 | 1, 27, 32, 49, 64, 81, 121, 125, 128, 135, ... = A379537
%e A379538 3 | 1, 32, 49, 64, 81, 98, 121, 125, 128, 169, ...
%e A379538 4 | 1, 27, 64, 81, 121, 125, 128, 169, 243, 256, ...
%e A379538 5 | 1, 27, 81, 125, 128, 162, 169, 243, 256, 289, ...
%e A379538 6 | 1, 81, 125, 243, 250, 256, 289, 343, 361, 375, ...
%e A379538 7 | 1, 64, 81, 125, 243, 343, 361, 375, 405, 486, ...
%e A379538 8 | 1, 64, 81, 125, 128, 243, 343, 512, 529, 567, ...
%e A379538 9 | 1, 81, 125, 128, 243, 256, 343, 625, 729, 768, ...
%e A379538 10 | 1, 125, 128, 243, 256, 343, 512, 625, 729, 1024, ... = A046759 (without the initial 1)
%e A379538 ... | \______ A379539 (main diagonal)
%e A379538 A377478
%e A379538 T(2,10) = 135 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); and 135 is the tenth number with this property.
%t A379538 Module[{dmax = 15, a, m}, a = Table[m = 0; Table[While[Total[IntegerLength[Select[Flatten[FactorInteger[++m]], # > 1 &], n]] >= IntegerLength[m, n]]; m, dmax-n+2], {n, dmax+1, 2, -1}]; Array[Diagonal[a, # - dmax] &, dmax]]
%Y A379538 Cf. A377478 (column k = 2), A379537 (row n = 2), A046759 (row n = 10), A379539 (main diagonal).
%Y A379538 Cf. A379373.
%K A379538 nonn,tabl,base
%O A379538 2,3
%A A379538 _Paolo Xausa_, Dec 25 2024