Wikipedia:Reference desk/Archives/Mathematics/2021 December 31

= December 31 =

Number theory
Except these 47 numbers:

{1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}

Do all positive integers which are not twice a square number can be written as (twice a positive square number) + (odd prime or twice an odd prime)?

Except these 8 numbers:

{1, 3, 4, 10, 14, 122, 422, 432}

Do all positive integers which are not twice a triangular number can be written as (twice a positive triangular number) + (odd prime or twice an odd prime)?

——114.41.123.50 (talk) 09:31, 31 December 2021 (UTC)


 * The first part - it seems likely that this is all of them. A quick and dirty program shows no more under 391,000,000 109.  Bubba73 You talkin' to me? 06:48, 1 January 2022 (UTC)


 * Like Goldbach's conjecture, these conjectures have a heuristic justification. They may also share resistance to proof attempts with Goldbach's conjecture, although the alleged proof of a weaker version inspires some hope. --Lambiam 12:48, 1 January 2022 (UTC)


 * And the second part, there are no others less than 185,000,000 109. Bubba73 You talkin' to me? 06:41, 2 January 2022 (UTC)


 * These are now sequences and .  If the person who asked the question will give their name, I will gladly give credit in the OEIS.  Bubba73 You talkin' to me? 22:47, 5 January 2022 (UTC)

Sum of reciprocals
The sum of reciprocals for “triangular numbers * k + 1” (where k is positive integer) is (see Centered_polygonal_number)


 * $$\frac{2\pi}{k\sqrt{1-\frac{8}{k}}}\tan\left(\frac{\pi}{2}\sqrt{1-\frac{8}{k}}\right)$$, if k ≠ 8


 * $$\frac{\pi^2}{8}$$, if k = 8

But what is the formula of the sum of reciprocals for “generalized pentagonal numbers * k + 1” (where k is positive integer)? (generalized pentagonal number is )

——114.41.123.50 (talk) 09:35, 31 December 2021 (UTC)


 * An observation. Just like the case $$k=8$$ is special for the case of triangular numbers, the case $$k=24$$ is special for pentagonal numbers: $$24 p_n + 1 = (6n - 1)^2.$$ --Lambiam 00:32, 1 January 2022 (UTC)