cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Dec 25 2024

Keywords

Comments

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).

Examples

			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).

Links

Crossrefs

Row n = 2 of A379538.
Cf. A046759, A070939, A377369, A379373.

Programs

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

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

Views

Author

Paolo Xausa, Dec 28 2024

Keywords

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.

Examples

			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.

Links

Crossrefs

Column k = 2 of A379538.
Cf. A046759, A379537.

Programs

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

A379539 a(n) is the (n-1)-st frugal number in base n.

Original entry on oeis.org

1, 32, 64, 125, 250, 343, 343, 625, 729, 1024, 1849, 2197, 2401, 3125, 4374, 5103, 6250, 7168, 8704, 9477, 11875, 13718, 15379, 17303, 20577, 22627, 24334, 27889, 30613, 32805, 36501, 39601, 45056, 50301, 53125, 59392, 63869, 69169, 75449, 78125, 85169, 89667, 94249
Offset: 2

Views

Author

Paolo Xausa, Dec 25 2024

Keywords

Comments

See A379538 for more information and links.

Crossrefs

Main diagonal of A379538.

Programs

  • Mathematica
    Module[{m}, Table[m = 0; Do[While[Total[IntegerLength[Select[Flatten[FactorInteger[++m]], # > 1 &], n]] >= IntegerLength[m, n]], n-1]; m, {n, 2, 45}]]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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