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).
125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715, 1792, 1849, 1875, 2048, 2187, 2197, 2209, 2401, 2560, 2809, 3125, 3481, 3584, 3645, 3721, 4096, 4374, 4375, 4489, 4802, 4913
Offset: 1
Examples
125 = 5^3, l(n) = 3 and D(n) = 2, so 125 is a member of the sequence.
References
- Bernardo Recamán, The Bogota Puzzles, Courier Dover Publications, Inc., 2020, p. 77.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
- C. K. Caldwell, The Prime Glossary, economical number.
- J.-M. De Koninck and F. Luca, On strings of consecutive economical numbers of arbitrary length, INTEGERS (2005), Volume: 5, Issue: 2, #A5.
- Jean-Paul Delahaye, Les chasseurs de nombres premiers, Pour la Science, No. 258 April 1999. [v1].
- R. G. E. Pinch, Economical numbers.
- R. G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
- W. Schneider, Economical Numbers.
- G. Villemin's Almanach of Numbers, Nombres Economes.
- Eric Weisstein's World of Mathematics, Economical Number.
- Wikipedia, Frugal number.
Programs
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Haskell
a046759 n = a046759_list !! (n-1) a046759_list = filter (\n -> a050252 n < a055642 n) [1..] -- Reinhard Zumkeller, Jun 21 2011
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Mathematica
ecoQ[n_] := Total[ Length /@ IntegerDigits /@ Flatten[ FactorInteger[n] /. {p_, 1} -> p]] < Length[ IntegerDigits[n]]; Select[ Range[5000], ecoQ] (* Jean-François Alcover, Jul 28 2011 *) -
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
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Python
from sympy import factorint 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)) print([k for k in range(5000) if ok(k)]) # Michael S. Branicky, Dec 22 2024
Extensions
More terms from Eric W. Weisstein
Comments