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

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

Author

Paolo Xausa, Dec 28 2024

Keywords

• nonn
• base

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

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

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}]]