A308416 Values of m for which 2*p + m cannot be a square when p is a prime.

1, 4, 8, 9, 13, 16, 17, 20, 24, 25, 28, 29, 33, 36, 37, 40, 41, 44, 48, 49, 52, 53, 56, 57, 61, 64, 65, 68, 69, 72, 73, 76, 80, 81, 84, 85, 88, 89, 92, 93, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 120, 121, 124, 125, 128, 129, 132, 133, 136, 137, 141, 144, 145, 148, 149

Offset: 1

Author

Bob Andriesse, May 25 2019

Keywords

• nonn

Comments

m = i^2 + 4*j is a term for i > 0, 0 <= j < i. Proof: If p = 2, then i^2 < 2*p + m < (i+2)^2. Therefore (i+1)^2 = 4 + i^2 + 4*j, which leads to a contradiction. If p > 2 is such that 2*p + i^2 + 4*j = k^2, then k + i and k - i are both even numbers. Therefore 4 | 2*p + 4*j, which is also a contradiction.

The terms of this sequence can be obtained by starting with A042948 (numbers congruent to 0 or 1 mod 4) and deleting the terms of A028347 (n^2 - 4).

Crossrefs

Cf. A042948, A028347.

Programs

Python

a=[]
a.append(0) #Offset starts at 1
iMax=15 #Example value
for i in range(1,iMax+1):
  for j in range(0,i):
   m=i*i+j*4
   a.append(m)
a.sort()

Formula

Conjecture: for k > 0 and 1 <= j <= k, a(2k^2-2j+1) = 4k^2+4k-4j-3, a(2k^2-2j+2) = 4k^2+4k-4j, a(2k^2+2k-2j+1) = 4k^2+8k-4j, a(2k^2+2k-2j+2) = 4k^2+8k-4j+1. - Jinyuan Wang, Jul 23 2019