A179312 - OEIS

%I #29 Nov 14 2014 08:18:02

%S 0,0,0,4,0,6,0,4,9,10,0,6,0,14,15,4,0,9,0,10,21,22,0,6,25,26,9,14,0,

%T 15,0,4,33,34,35,9,0,38,39,10,0,21,0,22,15,46,0,6,49,25,51,26,0,9,55,

%U 14,57,58,0,15,0,62,21,4,65,33,0,34,69,35,0,9,0,74

%N Largest semiprime dividing n, or 0 if no semiprime divides n.

%C a(p in primes A000040) = 0; a(k in semiprimes A001358) = k. This is to semiprimes A001358 as A006530 is to primes A000040.

%H Alois P. Heinz, <a href="/A179312/b179312.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = MAX(0, k in A001358 such that k | n).

%e The smallest semiprime is 4, so a(n<4) = 0.

%e a(4) = 4, since 4 = 2^2 is semiprime, and 4 | 4 (i.e. 4/4 = 1).

%e a(5) = 0 because 5 is prime, only 1 and 5 evenly divide 5, no prime (with 1 prime factor) is a semiprimes (with two prime factors, not necessarily distinct).

%e a(6) = 6, since 6 = 2*3 is semiprime, and 6 | ^ (i.e. 6/6 = 1).

%e a(8) = 4, since 4 = 2^2 is semiprime, and 4 | 8 (i.e. 8/4 = 2).

%p a:= proc(n) local l;

%p       if n<4 or isprime(n) then 0

%p     else l:= sort(ifactors(n)[2], (x, y)-> x[1]>y[1]);

%p          l[1][1] *l[`if`(l[1][2]>=2, 1, 2)][1]

%p       fi

%p     end:

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Jun 23 2012

%t semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; f[n_] := Max@ Select[ Divisors@ n, semiPrimeQ] /. {-\[Infinity] -> 0}; Array[f, 55]

%Y Cf. A001358, A006530, A034699, A052126, A052369, A061395

%Y Cf. A088739 (smallest semiprime divisor of n-th composite number)

%K nonn,easy

%O 1,4

%A _Jonathan Vos Post_, Jan 11 2011