A179939 Largest semiprime divisor of all composite numbers between semiprime(n) and semiprime(n+1), or 0 if there are none.

0, 4, 0, 6, 0, 10, 0, 6, 0, 15, 0, 0, 9, 0, 22, 6, 25, 26, 14, 0, 15, 21, 34, 35, 38, 39, 21, 0, 0, 22, 46, 0, 0, 51, 55, 57, 58, 0, 15, 0, 0, 62, 65, 0, 69, 0, 0, 9, 0, 77, 39, 0, 10, 82, 21, 87, 0, 91, 46, 93, 95, 65, 0, 0, 51, 0, 69, 106

Offset

1,2

Comments

This is to A052248 as semiprimes (A001358) are to primes (A000040). This defines a mapping f from semiprimes to semiprimes or 0 and f(s) < s holds for all semiprimes s. There is a block of k-1 consecutive 0's corresponding to each block of k consecutive semiprimes (i.e., a block of two consecutive 0's starting at the least of the triples in A115394).

Links

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

Formula

a(n) = max_{A001358(n) < k < A001358(n+1)} A179312(k).

Example

a(1) = 0 because there are no composite numbers between the 1st semiprime 4 and the 2nd semiprime 6.
a(2) = 4 because the composite numbers between the 2nd semiprime 6 and the 3rd semiprime 9 are {8} which is divisible by the semiprime 4=2*2.
a(10) = 15 because the composite numbers between the 10th semiprime 26 and the 11th semiprime 33 are {27, 28, 30, 32} of which the maximum is found for 30 which is divisible by the semiprime 15=3*5.

Maple

b:= proc(n) option remember; local k;
      if n=1 then 4
             else for k from b(n-1)+1 while
                    isprime(k) or add(i[2], i=ifactors(k)[2])<>2
                  do od; k
      fi
    end:
a:= proc(n) option remember; local k, l;
      k, l:= b(n)+1, b(n+1)-1;
      max(0, seq(seq(`if`(irem(j, b(i))=0, b(i), NULL),
                     i=1..n), j=k..l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 14 2011

Mathematica

(* First run the program for A105999 *) semiPrimeQ[x_] := TrueQ[Plus @@ Last /@ FactorInteger[x] == 2]; spGPF[start_, end_] := Module[{divList, spList}, divList = Union[Flatten[Table[Divisors[n], {n, start + 1, end - 1}]]]; spList = Select[divList, semiPrimeQ]; If[Length[spList] > 0, Return[Max[spList]], Return[0]]]; Table[spGPF[SemiPrime[n], SemiPrime[n + 1]], {n, 50}] (* Alonso del Arte, Jan 13 2011 *)

Crossrefs

Cf. A001358, A006530, A052248, A115394, A179312.

Sequence in context: A037282 A191558 A075083 * A163407 A023891 A075091

Adjacent sequences: A179936 A179937 A179938 * A179940 A179941 A179942

Keyword

nonn,easy

Author

Jonathan Vos Post, Jan 12 2011

Status

approved