login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A279070 Compact numbers: numbers that can be expressed more compactly using their prime factorization than their decimal expansion. 2
2187, 2401, 3125, 6561, 12167, 14641, 15625, 16384, 16807, 19683, 24389, 28561, 29791, 32768, 50653, 59049, 65536, 68921, 78125, 79507, 83521, 100489, 103823, 109375, 109561, 113569, 117649, 120409, 121801, 124609, 128881, 130321, 131072, 134689, 137781 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For any number k > 1, write its "compact prime factorization", with no spaces, as p1^e1*p2^e2*...*pj^ej, where p1, p2, ..., pj are the distinct prime factors of k and e1, e2, ..., ej are their respective exponents, but omit each exponent whose value is 1 (along with its caret character "^"). Sequence gives those numbers k whose compact prime factorization has fewer characters than k has decimal digits.

The smallest term other than a prime power is 109375 = 5^6*7.

The smallest term that is a power of 10 is 10000000 = 2^7*5^7.

The smallest term that is a factorial is 45!

= 119622220865480194561963161495657715064383733760000000000

= 2^41*3^21*5^10*7^6*11^4*13^3*17^2*19^2*23*29*31*37*41*43.

Includes 2^k for k >= 14, 3^k for k >= 7, 5^k for k >= 5, 7^k for k >= 4. - Robert Israel, Dec 26 2016

Let k'(b) be the smallest k such that b^k is included; then the sequence k'(2), k'(3), k'(4), ... begins {14, 7, 7, 5, 9, 4, 5, 4, 7, 4, 8, 4, 7, 6, 4, 4, 7, 4, 7, 6, 6, 3, 6, 3, ...} (with the larger values generally occurring where b has more than one prime divisor). It appears that b^k is included for all b > 1 and all k >= k'(b) with only two exceptions: although 6^k'(6) = 6^9 = 10077696 = 2^9*3^9 and 6^12 = 2176782336 = 2^12*3^12 are included, 6^10 = 60466176 = 2^10*3^10 and 6^11 = 362797056 = 2^11*3^11 are not. - Jon E. Schoenfield, Dec 26 2016

Note that there is another class of numbers that are called "Compact". See the definition in A169661. See also the links from T. M. Apostol and from V. Shevelev in the same entry. See also A070566 and A145554. - Omar E. Pol, Dec 26 2016

LINKS

Robert Israel, Table of n, a(n) for n = 1..1000

Jon E. Schoenfield, Magma program (outputs each term k with its compact prime factorization)

EXAMPLE

The number 2187 = 3^7 can be written more compactly as "3^7" (3 characters) than as "2187" (4 characters), so 2187 is in the sequence.

MAPLE

filter:=  proc(n) local F, t;

    F:= ifactors(n)[2];

    nops(F)-2+add(ilog10(t[1])+1+`if`(t[2]=1, 0, 2+ilog10(t[2])), t=F)<ilog10(n);

end proc:

select(filter, [$2..2*10^5]); # Robert Israel, Dec 26 2016

CROSSREFS

Cf. A070566, A079603, A145554.

Sequence in context: A152816 A321823 A215960 * A046320 A232926 A255113

Adjacent sequences:  A279067 A279068 A279069 * A279071 A279072 A279073

KEYWORD

nonn,base

AUTHOR

Jon E. Schoenfield, Dec 25 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 13 17:34 EST 2019. Contains 329106 sequences. (Running on oeis4.)