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

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

Offset: 1

Author

Jonathan Vos Post, Jan 11 2011

Keywords

• nonn
• easy

Comments

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

Examples

			The smallest semiprime is 4, so a(n<4) = 0.
a(4) = 4, since 4 = 2^2 is semiprime, and 4 | 4 (i.e., 4/4 = 1).
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).
a(6) = 6, since 6 = 2*3 is semiprime, and 6 | ^ (i.e., 6/6 = 1).
a(8) = 4, since 4 = 2^2 is semiprime, and 4 | 8 (i.e., 8/4 = 2).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

Programs

Maple

a:= proc(n) local l;
      if n<4 or isprime(n) then 0
    else l:= sort(ifactors(n)[2], (x, y)-> x[1]>y[1]);
         l[1][1] *l[`if`(l[1][2]>=2, 1, 2)][1]
      fi
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 23 2012

Mathematica

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

Formula

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