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

Sun Mar 26 14:22:20 CEST 2017

```Hi David,

Thank you for this message!

Please, at posing the problem do not
use a very rare abbreviation as
OGI-sequence. I till now do not
understand what did you mean
under that.

Best regards,

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of David Corneth [davidacorneth at gmail.com]
Sent: 25 March 2017 14:50
To: Sequence Fanatics Discussion list
Subject: [seqfan] A240751; a(n) is the smallest k such that in the prime power factorization of k! there exists at least one exponent n.

Hi all,

This sequence was submitted by Vladimir Shevelev and further
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

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