A064653 Integers not expressible as p + q*a^2, a>1 and p, q are primes.

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 16, 18, 24, 26, 28, 36, 42, 60, 72, 84, 90, 96, 108, 240, 300, 420, 1050, 1260

Offset: 1

Author

Robert G. Wilson v, Oct 07 2001

Keywords

• nonn
• more

Comments

Dean Hickerson (Oct 12 2001) writes: I suspect that there are no more terms in the sequence. In fact, I'll make the stronger conjecture that for all n>1260, n can be written as p + q*a^2 where a is the smallest prime that does not divide n. For example, for n=10080, a=11 and we have the representation 10080 = 7297 + 23 * 11^2.

There are no other terms up to 10^7.

Hickerson's stronger conjecture holds for n <= 10^9. Therefore, there are no other terms up to 10^9. - David A. Corneth, Jun 17 2019

Examples

			18 is in the sequence because p + 2*2^2 would imply that p is 10, or p + 2*3^2 would imply that p is 0, or p+ 3*2^2 would imply that p is 6, all of which are composite numbers.
		

Crossrefs

A subsequence of A064915.

Programs

Mathematica

Complement[Range[2000], Union@Flatten@Outer[Plus, Prime[Range[PrimePi[2000]]], Union@Flatten@Outer[Times, Prime[Range[PrimePi[2000]]], Table[a^2, {a, 2, 20}]]]] (* Robert Price, Jun 16 2019 *)

Extensions

Two more terms from Dean Hickerson, Oct 12 2001