A071621 Primes that can be written as "a * b + c * d", where a, b, c and d are also primes.

13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Author

Arnoud Buzing (arnoudb(AT)wolfram.com), Jun 21 2002

Keywords

• nonn

Comments

As size in the Mathematica coding is increased, the primes not previously covered will probably be forthcoming. Conjecture: Only the primes 2, 3, 5, 7, 11 and 17 are not representable by this form.

Examples

			13 = 2*2 + 3*3, so 13 belongs to the sequence.
		

Links

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

Programs

Mathematica

size = 15; Select[ Union[ Flatten[ Table[ ppp = Prime[i]Prime[j] + Prime[k]Prime[l]; If[ PrimeQ[ppp], Print[{Prime[i], Prime[j], Prime[k], Prime[l], ppp}]]; ppp, {i, size}, {j, size}, {k, size}, {l, size} ]]], PrimeQ]
Take[Select[Union[Total[Times@@@TakeDrop[#,2]]&/@Tuples[Prime[Range[15]],4]],PrimeQ],60] (* Harvey P. Dale, Dec 04 2024 *)

PARI

list(lim)=my(v=vectorsmall(lim\1),u=List(),t); forprime(p=3,lim\2-2, forprime(q=2,min(p,(lim-4)\p), t=p*q; forprime(r=2,(lim-t)\2, v[t+2*r]=1))); forprime(i=1,lim,if(v[i],listput(u,i))); v=0; Set(u) \\ Charles R Greathouse IV, Nov 05 2015

PARI

a(n)=if(n>1,prime(n+6),13) \\ Charles R Greathouse IV, Nov 05 2015

Extensions

Edited by Robert G. Wilson v, Jun 25 2002