A379373 -id:A379373 - cp's OEIS frontend

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

Paolo Xausa, Dec 25 2024

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).

			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).

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.

Row n = 2 of A379538.

Cf. A046759, A070939, A377369, A379373.

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

A379538 Square array read by ascending antidiagonals: T(n,k) is the k-th frugal number in base n.

Original entry on oeis.org

1, 1, 27, 1, 32, 32, 1, 27, 49, 49, 1, 27, 64, 64, 64, 1, 81, 81, 81, 81, 81, 1, 64, 125, 125, 121, 98, 121, 1, 64, 81, 243, 128, 125, 121, 125, 1, 81, 81, 125, 250, 162, 128, 125, 128, 1, 125, 125, 125, 243, 256, 169, 169, 128, 135, 1, 125, 128, 128, 128, 343, 289, 243, 243, 169, 147

Offset: 2

Paolo Xausa, Dec 25 2024

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).
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).

			Array begins:
  n\k| 1    2    3    4    5    6    7    8    9    10  ...
  ---------------------------------------------------------
   2 | 1,  27,  32,  49,  64,  81, 121, 125, 128,  135, ... = A379537
   3 | 1,  32,  49,  64,  81,  98, 121, 125, 128,  169, ...
   4 | 1,  27,  64,  81, 121, 125, 128, 169, 243,  256, ...
   5 | 1,  27,  81, 125, 128, 162, 169, 243, 256,  289, ...
   6 | 1,  81, 125, 243, 250, 256, 289, 343, 361,  375, ...
   7 | 1,  64,  81, 125, 243, 343, 361, 375, 405,  486, ...
   8 | 1,  64,  81, 125, 128, 243, 343, 512, 529,  567, ...
   9 | 1,  81, 125, 128, 243, 256, 343, 625, 729,  768, ...
  10 | 1, 125, 128, 243, 256, 343, 512, 625, 729, 1024, ... = A046759 (without the initial 1)
  ...       |                                         \______ A379539 (main diagonal)
         A377478
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.

Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
Giovanni Resta, Frugal numbers, Numbers Aplenty, 2013.
Wikipedia, Frugal number.

Cf. A377478 (column k = 2), A379537 (row n = 2), A046759 (row n = 10), A379539 (main diagonal).

Cf. A379373.

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]]

cp's OEIS Frontend

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

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