A238397 Numbers of the form pq + qr + rp where p, q and r are distinct primes (sorted sequence without duplicates).

31, 41, 59, 61, 71, 87, 91, 101, 103, 113, 119, 121, 129, 131, 143, 151, 161, 167, 171, 185, 191, 199, 211, 213, 215, 221, 227, 239, 241, 243, 247, 251, 263, 269, 271, 275, 281, 293, 297, 299, 301, 311, 321, 327, 331, 339, 341, 343, 347, 355

Offset: 1

Author

Jean-François Alcover, Feb 26 2014

Keywords

• nonn

Comments

Numbers of the form e2(p, q, r) for distinct primes p, q, r, where e2 is the elementary symmetric polynomial of degree 2. Other sequences are obtained with different numbers of distinct primes and degrees: A000040 for 1 prime, A038609 and A006881 for 2 primes, A124867, this sequence, and A007304 for 3 primes. The 4-prime sequences are not presently in the OEIS with the exception of A046386. - Charles R Greathouse IV, Feb 26 2014

Examples

			71 = 3*5 + 5*7 + 7*3 = 2*3 + 3*13 + 13*2 is in the sequence (only once, though 2 solutions exist).
		

Links

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

Crossrefs

Cf. A087053, A087054.

Programs

Mathematica

terms = 50; dm (* initial number of primes *) = 10; f[p_, q_, r_] := p*q + q*r + r*p; Clear[A238397]; A238397[m_] := A238397[m] = Take[u = Union[f @@@ Subsets[Prime /@ Range[m], {3}]], Min[Length[u], terms]]; A238397[dm]; A238397[m = 2*dm]; While[Print["m = ", m]; A238397[m] != A238397[m - dm], m = m + dm]; A238397[m]

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(1)))); 0 \\ Charles R Greathouse IV, Feb 26 2014

PARI

list(n)=my(v=List()); 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), listput(v, p*S+P)))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014