A379537 -id:A379537 - cp's OEIS frontend

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

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]]

A377478 a(n) = first frugal number > 1 in base n.

Original entry on oeis.org

27, 32, 27, 27, 81, 64, 64, 81, 125, 125, 243, 243, 243, 243, 256, 343, 343, 512, 512, 512, 512, 625, 625, 625, 729, 729, 1024, 1024, 1024, 1024, 1024, 1331, 1331, 1331, 1331, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2187, 2401, 2401, 2401, 3125, 3125

Offset: 2

Paolo Xausa, Dec 28 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.

			a(2) = 27 because 27 = 3^3 = 11_2^11_2; the total number of bits of (11_2, 11_2) = 4 < the number of bits of 27 = 11011_2 (5); and 27 is the first number > 1 with this property.
a(3) = 32 because 32 = 2^5 = 2_3^12_3; the total number of digits of (2_3, 12_3) = 3 < the number of digits in base 3 of 32 = 1012_3 (4); and 32 is the first number > 1 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.

Column k = 2 of A379538.

Cf. A046759, A379537.

Module[{m}, Table[m = 1; While[Total[IntegerLength[Select[Flatten[FactorInteger[++m]], # > 1 &], n]] >= IntegerLength[m, n]]; m, {n, 2, 50}]]

cp's OEIS Frontend

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

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