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

1,71

Comments

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

Links

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

Example

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

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}]

Prog

(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

Crossrefs

Cf. A238403, A087053, A087054.

Sequence in context: A281669 A331845 A014083 * A112315 A295663 A005926

Adjacent sequences: A238400 A238401 A238402 * A238404 A238405 A238406

Keyword

nonn

Author

Jean-François Alcover, Feb 26 2014

Status

approved