A238403 Number of ways a number can be decomposed as a sum of the form pq + qr + rp where p < q < r are distinct primes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Jean-François Alcover, Feb 26 2014

Keywords

• nonn

Comments

The average value of a(n) is >> sqrt(n)/log^3 n. - Charles R Greathouse IV, Feb 26 2014

Examples

			71 = 3*5 + 5*7 + 7*3 = 2*3 + 3*13 + 13*2, therefore a(71) = 2.
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A238403, A087053, A087054.

Programs

Mathematica

r[n_, p_] := Reduce[p < q < r && p*q+q*r+r*p == n, {q, r}, Primes]; a[n_] := (For[cnt = 0; p = 2, p <= Ceiling[(n-6)/5], p = NextPrime[p], rnp = r[n, p]; If[rnp =!= False, Which[rnp[[0]] === And, Print["n = ", n, " ", {p, q, r} /. ToRules[rnp]]; cnt++, rnp[[0]] === Or, Print["n = ", n, " ", {p, q, r} /. {ToRules[rnp]}]; cnt += Length[rnp], True, Print["error: n = ", n, " ", rnp]]]]; cnt); Table[a[n], {n, 1, 100}]

PARI

list(n)=my(v=vector(n)); forprime(r=5,(n-6)\5, forprime(q=3, min((n-2*r)\(r+2),r-2), my(S=q+r,P=q*r); forprime(p=2,min((n-P)\S,q-1), v[p*S+P]++))); v \\ Charles R Greathouse IV, Feb 26 2014

PARI

a(n)=my(s);forprime(r=(sqrtint(3*n-3)+5)\3,(n-6)\5, forprime(q= sqrtint(r^2+n)-r+1,min((n-2*r)\(r+2),r-2),if((n-q*r)%(q+r)==0 && isprime((n-q*r)/(q+r)),s++)));s \\ Charles R Greathouse IV, Feb 26 2014