A379373 Numbers k such that A050252(k) <= A055642(k).

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 105, 106, 107, 109, 111, 112, 113, 115, 118, 119, 121, 122, 123, 125, 127, 128, 129, 131, 133, 134, 135

Offset: 1

Author

Paolo Xausa, Dec 21 2024

Keywords

• nonn
• base

Comments

Some authors call these numbers "economical numbers" (see links) and refer to A046759 as "frugal numbers", while other authors define "economical numbers" to be A046759.

First differs from A046758 at n = 54, where a(54) = 125 is missing from A046758.

Examples

			112 is a term because 112 = (2^4)*7; the total number of digits of (2, 4, 7) = 1 + 1 + 1 <= the number of digits of 112 (3).
125 is a term because 125 = 5^3; the total number of digits of (5, 3) = 1 + 1 <= the number of digits of 125 (3).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
• Giovanni Resta, Economical numbers, Numbers Aplenty, 2013.

Crossrefs

Union of A046758 and A046759.

Complement of A046760.

Cf. A050252, A055642.

Programs

Mathematica

A379373Q[k_] := Total[IntegerLength[Select[Flatten[FactorInteger[k]], # > 1 &]]] <= IntegerLength[k];
Select[Range[200], A379373Q]

Python

from sympy import factorint
def ok(n): return n > 0 and sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items()) <= len(str(n))
print([k for k in range(136) if ok(k)]) # Michael S. Branicky, Dec 22 2024