[seqfan] A240751; a(n) is the smallest k such that in the prime power factorization of k! there exists at least one exponent n.

[David Corneth]

Hi all,

This sequence was submitted by Vladimir Shevelev and further
researched by Vladimir
Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses.

Following those sequences I submitted A284050 and A284051 which are
respectively a(n) = floor(A240751(n)/n) and a(n) = A240751(n) mod n.
They're interesting, I think, because A284050 illustrates how to find
A240751(n) and both sequences have surprisingly low terms.
I find A240751 intriguing and it begs some questions.

Let a(n) is the least term k such that prime(n)^k || A240751(k)!
First few terms are 1, 2, 12, 29, 186, 2865, 3265, 379852, 7172525.
Is this sequence infinite?

What are values n such that p^n||A240751(n) for some prime p?
As primes get larger, the density seems to decrease.
For p = 2 we have 1, 3, 4, 7, 8, 10, 11, 15, 16, 18, 19, 22, 23, 25, 26,
31, 32, 34, 35, 38, 39, ...

For p = 3 we have 2, 5, 6, 9, 13, 14, 17, 21, 27, 28, 30, 36, 40, 44, 45,
48, 55, 58, 59, 61, 62, ...

For p = 5 we have 12, 20, 24, 33, 37, 43, 51, 52, 65, 69, 77, 83, 87, 96,
100, 118, 124, 132, 133, 160, ...

For p = 7 we have 29, 60, 91, 92, 141, 154, 185, 204, 217, 241, 254, 279,
285, 342, 403, 441, 473, 497, 510, 528, ...

For p = 11 we have 186, 447, 635, 765, 1035, 1092, 1378, 1435, 1540, 2015,
2553, 2740, 2808, 3027, 4154, 4465, 4497, 4603, 4766, 4816, ...

For p = 13 we have 2865, 3640, 3942, 4922, 6677, 7959, 10972, 11577, 11928,
12859, 18233, 19213, 22153, 30295, 30646, 31977, 34664, 35620, 36527,
44440, ...

For p = 17 we have 3265, 5322, 7209, 15545, 56020, 79639, 90989, 176208,
183615, 198389, 256201, 263608, 287226, 329758, 362670, 400301, 408639,
409570, 448157, 454945, ...

For p = 19 we have 379852, 1937399, 3213587, 3693182, 3929909, 5297908,
5491483, 5558857, 5585137, 6073350, 6098602, 6239688, 6456726, 6621601,
7628332, 7676090, 7784700, ... (no more below 10^7)

For p = 23 we have 7172525, 15122988, 21210412, 48538612, 48752905,
51466002, 52658325, 55938150, 57611786, 73564837, 79352750, 83297618,
87015476, 96850742, ... (no more below 10^8)

Below 10^8 there are no terms for p > 23. The union of these sequences seem
to give the positive integers.

Are some of these sequences OGI?
Any thoughts or ideas?
Best,
David