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

1,1

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

Links

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

Example

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

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 *)

Prog

(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)

Crossrefs

Cf. A238397.

Sequence in context: A050672 A122531 A141871 * A050806 A243890 A129563

Adjacent sequences: A238500 A238501 A238502 * A238504 A238505 A238506

Keyword

nonn

Author

Charles R Greathouse IV, Feb 27 2014

Status

approved