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

2,3

Comments

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).

Links

Table of n, a(n) for n=2..67.

Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.

Giovanni Resta, Frugal numbers, Numbers Aplenty, 2013.

Wikipedia, Frugal number.

Example

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.

Mathematica

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]]

Crossrefs

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

Cf. A379373.

Sequence in context: A040754 A059952 A244110 * A338659 A040755 A261197

Adjacent sequences: A379535 A379536 A379537 * A379539 A379540 A379541

Keyword

nonn,tabl,base

Author

Paolo Xausa, Dec 25 2024

Status

approved