A238503 Numbers of the form pq + pr + ps + qr + qs + rs where p, q, r, and s are distinct primes.

101, 141, 161, 173, 197, 201, 213, 221, 236, 241, 245, 261, 266, 269, 297, 317, 321, 325, 326, 333, 341, 350, 356, 365, 373, 377, 381, 389, 393, 401, 404, 413, 416, 426, 429, 441, 453, 461, 464, 465, 466, 476, 481, 485, 488, 493, 501, 505, 506

Offset: 1

Author

Charles R Greathouse IV, Feb 27 2014

Keywords

• nonn

Comments

Numbers of the form e2(p, q, r, s) for distinct primes p, q, r, s 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, A238397, and A007304 for 3 primes.

What is the density of this sequence, and is it less than 1? There are 701917 terms below a million and 7042080 below 10^7.

There are 70307093 terms below 10^8. - Charles R Greathouse IV, Jun 14 2017

Examples

			101 = 2*3 + 2*5 + 2*7 + 3*5 + 3*7 + 5*7.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A238397.

Programs

Mathematica

pqrs[{p_,q_,r_,s_}]:=Total[Times@@@Subsets[{p,q,r,s},{2}]]; Take[ Flatten[ pqrs/@Subsets[Prime[Range[20]],{4}]]//Union,50] (* Harvey P. Dale, Jan 17 2021 *)

PARI

list(n)=my(v=List()); forprime(s=7,(n-31)\10,forprime(r=5, min((n-6-5*s)\(s+5),s-2), forprime(q=3, min((n-2*r-2*s-r*s)\(s+r+2), r-2), my(S=q+r+s, P=q*r+r*s+q*s); forprime(p=2, min((n-P)\S, q-1), listput(v, p*S+P))))); Set(v)

PARI

list(n)=my(v=vectorsmall(n),u=List()); forprime(s=7,(n-31)\10,forprime(r=5, min((n-6-5*s)\(s+5),s-2), forprime(q=3, min((n-2*r-2*s-r*s)\(s+r+2), r-2), my(S=q+r+s, P=q*r+r*s+q*s); forprime(p=2, min((n-P)\S, q-1), v[p*S+P]=1)))); for(i=1,n,if(v[i],listput(u,i))); Vec(u)