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A046758 Equidigital numbers.

%I A046758 #55 Feb 16 2025 08:32:39
%S A046758 1,2,3,5,7,10,11,13,14,15,16,17,19,21,23,25,27,29,31,32,35,37,41,43,
%T A046758 47,49,53,59,61,64,67,71,73,79,81,83,89,97,101,103,105,106,107,109,
%U A046758 111,112,113,115,118,119,121,122,123,127,129,131,133,134,135,137,139
%N A046758 Equidigital numbers.
%C A046758 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).
%C A046758 The term "equidigital number" was coined by Recamán (1995). - _Amiram Eldar_, Mar 10 2024
%D A046758 Bernardo Recamán Santos, Equidigital representation: problem 2204, J. Rec. Math., Vol. 27, No. 1 (1995), pp. 58-59.
%H A046758 Reinhard Zumkeller, <a href="/A046758/b046758.txt">Table of n, a(n) for n = 1..10000</a>
%H A046758 J. P. Delahaye, "Primes Hunters", <a href="https://web.archive.org/web/20020703182514/http://www.pour-la-science.com/numeros/pls-258/logique.htm#int5">Economical and Prodigal Numbers (Text in French)</a>. [Wayback Machine link]
%H A046758 R. G. E. Pinch, <a href="https://arxiv.org/abs/math/9802046">Economical numbers</a>, arXiv:math/9802046 [math.NT], 1998.
%H A046758 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EquidigitalNumber.html">Equidigital Number.</a>.
%H A046758 Wikipedia, <a href="http://en.wikipedia.org/wiki/Equidigital_number">Equidigital number</a>.
%F A046758 A050252(a(n)) = A055642(a(n)). - _Reinhard Zumkeller_, Jun 21 2011
%e A046758 For n = 125 = 5^3, l(n) = 3 but D(n) = 2. So 125 is not a member of this sequence.
%t A046758 edQ[n_] := Total[IntegerLength[DeleteCases[Flatten[FactorInteger[n]], 1]]] == IntegerLength[n]; Join[{1}, Select[Range[140], edQ]] (* _Jayanta Basu_, Jun 28 2013 *)
%o A046758 (Haskell)
%o A046758 a046758 n = a046758_list !! (n-1)
%o A046758 a046758_list = filter (\n -> a050252 n == a055642 n) [1..]
%o A046758 -- _Reinhard Zumkeller_, Jun 21 2011
%o A046758 (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
%o A046758 (Python)
%o A046758 from itertools import count, islice
%o A046758 from sympy import factorint
%o A046758 def A046758_gen(): # generator of terms
%o A046758     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()))
%o A046758 A046758_list = list(islice(A046758_gen(),20)) # _Chai Wah Wu_, Feb 18 2022
%Y A046758 Cf. A046759, A046760, A050252, A055642, A073048.
%K A046758 nonn,base,easy
%O A046758 1,2
%A A046758 _N. J. A. Sloane_
%E A046758 More terms from _Eric W. Weisstein_