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

1,4

Comments

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

Links

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

Formula

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

Example

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).

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]

Crossrefs

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

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

Sequence in context: A016681 A210625 A210615 * A076290 A198224 A178105

Adjacent sequences: A179309 A179310 A179311 * A179313 A179314 A179315

Keyword

nonn,easy

Author

Jonathan Vos Post, Jan 11 2011

Status

approved