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

1,2

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

Links

Table of n, a(n) for n=1..28.

Example

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.

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 *)

Crossrefs

A subsequence of A064915.

Sequence in context: A115569 A343682 A343744 * A130588 A079238 A079042

Adjacent sequences: A064650 A064651 A064652 * A064654 A064655 A064656

Keyword

nonn,more

Author

Robert G. Wilson v, Oct 07 2001

Extensions

Two more terms from Dean Hickerson, Oct 12 2001

Status

approved