A087054 Primes of the form pq + qr + rp where p, q and r are distinct primes.

31, 41, 59, 61, 71, 101, 103, 113, 131, 151, 167, 191, 199, 211, 227, 239, 241, 251, 263, 269, 271, 281, 293, 311, 331, 347, 359, 383, 401, 419, 421, 431, 439, 461, 467, 479, 487, 491, 503, 521, 541, 563, 571, 587, 599, 607, 617, 631, 641, 647, 653, 661, 691

Offset

1,1

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Example

A003415(2*3*19)=2*3+3*19+19*2=101=A000040(26), therefore 101 is a term (but also A003415(2*5*13)=2*5+5*13+13*2=101).

Mathematica

sumProd[p_, q_, r_]:=p*q+p*r+q*r; pqrPrimes[nn_] := Module[{p=Prime[Range[PrimePi[(nn-6)/5]+1]], i, j, k, n}, Union[Reap[i=0; While[i++; sumProd[p[[i]], p[[i+1]], p[[i+2]]] <= nn, j=i; While[j++; sumProd[p[[i]], p[[j]], p[[j+1]]] <= nn, k=j; While[k++; n=sumProd[p[[i]], p[[j]], p[[k]]]; n <= nn, If[PrimeQ[n], Sow[n]]]]]][[2, 1]]]]; pqrPrimes[1000] (* T. D. Noe, Apr 27 2011 *)
nn=100; Take[Select[Union[Total[Times@@@Subsets[#, {2}]]&/@Subsets[ Prime[ Range[ nn]], {3}]], PrimeQ], nn] (* Harvey P. Dale, Jan 08 2013 *)

Prog

(PARI) list(lim)=my(v=List()); forprime(r=5, (lim-6)\5, forprime(q=3, min((lim-2*r)\(r+2), r-2), my(S=q+r, P=q*r); forprime(p=2, min((lim-P)\S, q-1), isprime(p*S+P) && listput(v, p*S+P)))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014
(PARI) is(n)=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)), return(isprime(n))))); 0 \\ Charles R Greathouse IV, Feb 26 2014

Crossrefs

Cf. A087053 (numbers of the form pq+qr+rp).

Cf. A189759 (p*q*r for primes of this form).

Sequence in context: A104822 A198175 A238397 * A245650 A363187 A098711

Adjacent sequences: A087051 A087052 A087053 * A087055 A087056 A087057

Keyword

nonn

Author

Reinhard Zumkeller, Aug 07 2003

Extensions

Corrected by T. D. Noe, Apr 27 2011

Status

approved