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

%I

%S 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,

%T 420,1050,1260

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

%C _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.

%C There are no other terms up to 10^7.

%C 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

%e 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.

%t 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 *)

%Y A subsequence of A064915.

%K nonn,more

%O 1,2

%A _Robert G. Wilson v_, Oct 07 2001

%E Two more terms from _Dean Hickerson_, Oct 12 2001

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Last modified January 22 21:47 EST 2022. Contains 350504 sequences. (Running on oeis4.)