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A179312 Largest semiprime dividing n, or 0 if no semiprime divides n.

%I A179312 #31 Jul 22 2025 21:26:21
%S A179312 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 A179312 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 A179312 14,57,58,0,15,0,62,21,4,65,33,0,34,69,35,0,9,0,74
%N A179312 Largest semiprime dividing n, or 0 if no semiprime divides n.
%C A179312 a(p in primes A000040) = 0; a(k in semiprimes A001358) = k. This is to semiprimes A001358 as A006530 is to primes A000040.
%H A179312 Alois P. Heinz, <a href="/A179312/b179312.txt">Table of n, a(n) for n = 1..10000</a>
%F A179312 a(n) = MAX(0, k in A001358 such that k | n).
%e A179312 The smallest semiprime is 4, so a(n<4) = 0.
%e A179312 a(4) = 4, since 4 = 2^2 is semiprime, and 4 | 4 (i.e., 4/4 = 1).
%e A179312 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 A179312 a(6) = 6, since 6 = 2*3 is semiprime, and 6 | ^ (i.e., 6/6 = 1).
%e A179312 a(8) = 4, since 4 = 2^2 is semiprime, and 4 | 8 (i.e., 8/4 = 2).
%p A179312 a:= proc(n) local l;
%p A179312       if n<4 or isprime(n) then 0
%p A179312     else l:= sort(ifactors(n)[2], (x, y)-> x[1]>y[1]);
%p A179312          l[1][1] *l[`if`(l[1][2]>=2, 1, 2)][1]
%p A179312       fi
%p A179312     end:
%p A179312 seq(a(n), n=1..80);  # _Alois P. Heinz_, Jun 23 2012
%t A179312 semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; f[n_] := Max@ Select[ Divisors@ n, semiPrimeQ] /. {-\[Infinity] -> 0}; Array[f, 55]
%Y A179312 Cf. A001358, A006530, A034699, A052126, A052369, A061395.
%Y A179312 Cf. A088739 (smallest semiprime divisor of n-th composite number).
%K A179312 nonn,easy
%O A179312 1,4
%A A179312 _Jonathan Vos Post_, Jan 11 2011