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A118133 Least concatenation of n to have exactly n prime factors with multiplicity. 1

%I #11 Mar 04 2018 03:16:33

%S 11,22,333,444,55555555,66666666,777777777777777,888888,999999999999,

%T 101010101010101010,11111111111111111111111111111111,

%U 121212121212121212121212,131313131313131313131313131313,141414141414141414141414141414

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

%C 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.

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

%e a(1) = 11 has 1 prime factor.

%e a(2) = 22 = 2 * 11 has 2 prime factors.

%e a(3) = 333 = 3^2 * 37 has 3 prime factors.

%e a(4) = 444 = 2^2 * 3 * 37 has 4 prime factors.

%e a(5) = 55555555 = 5 * 11 * 73 * 101 * 137 has 5 prime factors.

%e a(6) = 66666666 = 2 * 3 * 11 * 73 * 101 * 137 has 6 prime factors.

%e a(7) = 777777777777777 = 3 * 7 * 31 * 37 * 41 * 271 * 2906161.

%e a(8) = 888888 = 2^3 * 3 * 7 * 11 * 13 * 37.

%e a(9) = 999999999999 = 3^3 * 7 * 11 * 13 * 37 * 101 * 9901.

%e a(10) = 101010101010101010 = 2 * 3^2 * 5 * 7 * 13 * 19 * 37 * 52579 * 333667.

%e a(11) = 11111111111111111111111111111111 = 11 * 17 * 73 * 101 * 137 * 353 * 449 * 641 * 1409 * 69857 * 5882353.

%e a(12) = 121212121212121212121212 = 2^2 * 3^2 * 7 * 13 * 37 * 73 * 101 * 137 * 9901 * 99990001.

%t 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 *)

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

%Y Cf. A000461, A001222.

%K base,nonn

%O 1,1

%A _Jonathan Vos Post_, May 13 2006

%E More terms from _Harvey P. Dale_, Sep 07 2014

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)