A118133 Least concatenation of n to have exactly n prime factors with multiplicity.

11, 22, 333, 444, 55555555, 66666666, 777777777777777, 888888, 999999999999, 101010101010101010, 11111111111111111111111111111111, 121212121212121212121212, 131313131313131313131313131313, 141414141414141414141414141414

Offset

1,1

Comments

See also A000461 Concatenate n n times. Bigomega(n) = A001222 Number of prime divisors of n (counted with multiplicity). The numbers k of concatenations on n in a(n) are: 2, 2, 3, 3, 8, 8, 15, 6, 12, 9, 16, 12.

Links

Table of n, a(n) for n=1..14.

Formula

a(n) = nnn...nnn (k times) such that A001222(a(n)) = n, for k minimum.

Example

a(1) = 11 has 1 prime factor.
a(2) = 22 = 2 * 11 has 2 prime factors.
a(3) = 333 = 3^2 * 37 has 3 prime factors.
a(4) = 444 = 2^2 * 3 * 37 has 4 prime factors.
a(5) = 55555555 = 5 * 11 * 73 * 101 * 137 has 5 prime factors.
a(6) = 66666666 = 2 * 3 * 11 * 73 * 101 * 137 has 6 prime factors.
a(7) = 777777777777777 = 3 * 7 * 31 * 37 * 41 * 271 * 2906161.
a(8) = 888888 = 2^3 * 3 * 7 * 11 * 13 * 37.
a(9) = 999999999999 = 3^3 * 7 * 11 * 13 * 37 * 101 * 9901.
a(10) = 101010101010101010 = 2 * 3^2 * 5 * 7 * 13 * 19 * 37 * 52579 * 333667.
a(11) = 11111111111111111111111111111111 = 11 * 17 * 73 * 101 * 137 * 353 * 449 * 641 * 1409 * 69857 * 5882353.
a(12) = 121212121212121212121212 = 2^2 * 3^2 * 7 * 13 * 37 * 73 * 101 * 137 * 9901 * 99990001.

Mathematica

Table[Module[{i=1}, While[PrimeOmega[FromDigits[PadRight[{}, i, IntegerDigits[ n]]]]!= n, i++]; FromDigits[PadRight[{}, i, IntegerDigits[ n]]]], {n, 15}] (* Harvey P. Dale, Sep 07 2014 *)

Prog

(PARI) A118133(n)={local(r); r=n; while(bigomega(r)<>n, r=eval(Str(r, n))); r} \\ Michael B. Porter, May 01 2010

Crossrefs

Cf. A000461, A001222.

Sequence in context: A067894 A094620 A077431 * A345404 A225186 A155973

Adjacent sequences: A118130 A118131 A118132 * A118134 A118135 A118136

Keyword

base,nonn

Author

Jonathan Vos Post, May 13 2006

Extensions

More terms from Harvey P. Dale, Sep 07 2014

Status

approved