A046758 Equidigital numbers.

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, 127, 129, 131, 133, 134, 135, 137, 139

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Comments

Write n as product of primes raised to powers, let D(n) = A050252 = total number of digits in product representation (number of digits in all the primes plus number of digits in all the exponents that are greater than 1) and l(n) = number of digits in n; sequence gives n such that D(n)=l(n).

The term "equidigital number" was coined by Recamán (1995). - Amiram Eldar, Mar 10 2024

Examples

			For n = 125 = 5^3, l(n) = 3 but D(n) = 2. So 125 is not a member of this sequence.
		

References

• Bernardo Recamán Santos, Equidigital representation: problem 2204, J. Rec. Math., Vol. 27, No. 1 (1995), pp. 58-59.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• J. P. Delahaye, "Primes Hunters", Economical and Prodigal Numbers (Text in French). [Wayback Machine link]
• R. G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
• Eric Weisstein's World of Mathematics, Equidigital Number..
• Wikipedia, Equidigital number.

Crossrefs

Cf. A046759, A046760, A050252, A055642, A073048.

Programs

Haskell

a046758 n = a046758_list !! (n-1)
a046758_list = filter (\n -> a050252 n == a055642 n) [1..]
-- Reinhard Zumkeller, Jun 21 2011

Mathematica

edQ[n_] := Total[IntegerLength[DeleteCases[Flatten[FactorInteger[n]], 1]]] == IntegerLength[n]; Join[{1}, Select[Range[140], edQ]] (* Jayanta Basu, Jun 28 2013 *)

PARI

for(n=1, 100, s=""; F=factor(n); for(i=1, #F[, 1], s=concat(s, Str(F[i, 1])); s=concat(s, Str(F[i, 2]))); c=0; for(j=1, #F[, 2], if(F[j, 2]==1, c++)); if(#digits(n)==#s-c, print1(n, ", "))) \\ Derek Orr, Jan 30 2015

Python

from itertools import count, islice
from sympy import factorint
def A046758_gen(): # generator of terms
    return (n for n in count(1) if n == 1 or len(str(n)) == sum(len(str(p))+(len(str(e)) if e > 1 else 0) for p, e in factorint(n).items()))
A046758_list = list(islice(A046758_gen(),20)) # Chai Wah Wu, Feb 18 2022

Formula

A050252(a(n)) = A055642(a(n)). - Reinhard Zumkeller, Jun 21 2011

Extensions

More terms from Eric W. Weisstein