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A046759 Economical numbers: write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n) < l(n).

%I A046759 #78 Feb 16 2025 08:32:39
%S A046759 125,128,243,256,343,512,625,729,1024,1029,1215,1250,1280,1331,1369,
%T A046759 1458,1536,1681,1701,1715,1792,1849,1875,2048,2187,2197,2209,2401,
%U A046759 2560,2809,3125,3481,3584,3645,3721,4096,4374,4375,4489,4802,4913
%N A046759 Economical numbers: write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n) < l(n).
%C A046759 A050252(a(n)) < A055642(a(n)). - _Reinhard Zumkeller_, Jun 21 2011
%D A046759 Bernardo Recamán, The Bogota Puzzles, Courier Dover Publications, Inc., 2020, p. 77.
%H A046759 Reinhard Zumkeller, <a href="/A046759/b046759.txt">Table of n, a(n) for n = 1..10000</a>
%H A046759 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/EconomicalNumber.html">economical number</a>.
%H A046759 J.-M. De Koninck and F. Luca, <a href="http://www.emis.de/journals/INTEGERS/papers/a5int2003/a5int2003.Abstract.html">On strings of consecutive economical numbers of arbitrary length</a>, INTEGERS (2005), Volume: 5, Issue: 2, #A5.
%H A046759 Jean-Paul Delahaye, <a href="https://www.pourlascience.fr/sd/evolution/pour-la-science-n258-455.php">Les chasseurs de nombres premiers</a>, Pour la Science, No. 258 April 1999. [<a href="https://www.cristal.univ-lille.fr/~jdelahay/dnalor/ChasseursNbPr.pdf">v1</a>].
%H A046759 R. G. E. Pinch, <a href="http://www.s369624816.websitehome.co.uk/rgep/rpub.html#62">Economical numbers</a>.
%H A046759 R. G. E. Pinch, <a href="https://arxiv.org/abs/math/9802046">Economical numbers</a>, arXiv:math/9802046 [math.NT], 1998.
%H A046759 W. Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/economical-numbers.html">Economical Numbers</a>.
%H A046759 G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Decompos/Econome.htm">Nombres Economes</a>.
%H A046759 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EconomicalNumber.html">Economical Number.</a>
%H A046759 Wikipedia, <a href="http://en.wikipedia.org/wiki/Frugal_number">Frugal number</a>.
%e A046759 125 = 5^3, l(n) = 3 and D(n) = 2, so 125 is a member of the sequence.
%t A046759 ecoQ[n_] := Total[ Length /@ IntegerDigits /@ Flatten[ FactorInteger[n] /. {p_, 1} -> p]] < Length[ IntegerDigits[n]]; Select[ Range[5000], ecoQ] (* _Jean-François Alcover_, Jul 28 2011 *)
%o A046759 (Haskell)
%o A046759 a046759 n = a046759_list !! (n-1)
%o A046759 a046759_list = filter (\n -> a050252 n < a055642 n) [1..]
%o A046759 -- _Reinhard Zumkeller_, Jun 21 2011
%o A046759 (PARI) is(n)=my(f=factor(n));sum(i=1,#f[,1], #Str(f[i,1])+if(f[i,2]>1, #Str(f[i,2])))<#Str(n) && n>1 \\ _Charles R Greathouse IV_, Feb 01 2013
%o A046759 (Python)
%o A046759 from sympy import factorint
%o A046759 def ok(n): return n > 1 and sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items()) < len(str(n))
%o A046759 print([k for k in range(5000) if ok(k)]) # _Michael S. Branicky_, Dec 22 2024
%Y A046759 Cf. A046758, A046760.
%K A046759 nonn,nice,base
%O A046759 1,1
%A A046759 _N. J. A. Sloane_, Dec 11 1999
%E A046759 More terms from _Eric W. Weisstein_