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

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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 (sequence A347567 in the OEIS) and (sequence A347568 in the OEIS). 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#Sum_of_Reciprocals)

${\frac {2\pi }{k{\sqrt {1-{\frac {8}{k}}}}}}\tan \left({\frac {\pi }{2}}{\sqrt {1-{\frac {8}{k}}}}\right)$, 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 OEIS: A001318)

——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: $24p_{n}+1=(6n-1)^{2}.$  --Lambiam 00:32, 1 January 2022 (UTC)