Numbers not Sum of Square and Prime

Sequence

The sequence of (strictly) positive integers which is not the sum of a square and a prime begins:

$1, 10, 25, 34, 58, 64, 85, 91, 121, 130, 169, 196, \ldots$

This sequence is A014090 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

$1$ is trivial.

No prime number is in this sequence, as trivially:

$p = p + 0^2$

and so is the sum of a prime (itself), and $0^2$, which is square.

From Square of n such that 2n-1 is Composite is not Sum of Square and Prime, the sequence contains all $n^2$ such that $2 n - 1$ is composite.

The sequence of such $n$ is infinite, and begins::

$1, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 61, 62, 63, 65, \ldots$

This sequence is A104275 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The sequence of remaining integers is:

$10, 34, 58, 85, 91, 130, 214, 226, 370, 526, 706, 730, 771, 1255, 1351, 1414, 1906, 2986, 3676, 9634, 21679$

This sequence is A020495 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

From Non-Square Positive Integers not Sum of Square and Prime, this sequence is conjectured to be complete.

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10$