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

1,1

Comments

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

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.

J.-P. Delahaye, Les chasseurs de nombres premiers [broken link]

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

Example

125 = 5^3, l(n) = 3 and D(n) = 2, so 125 is a member of the sequence.

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 *)

Prog

(Haskell)
a046759 n = a046759_list !! (n-1)
a046759_list = filter (\n -> a050252 n < a055642 n) [1..]
-- Reinhard Zumkeller, Jun 21 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

Crossrefs

Cf. A046758, A046760.

Sequence in context: A195420 A080538 A104233 * A115938 A126895 A202240

Adjacent sequences: A046756 A046757 A046758 * A046760 A046761 A046762

Keyword

nonn,nice,base

Author

N. J. A. Sloane, Dec 11 1999

Extensions

More terms from Eric W. Weisstein

Status

approved