The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A131657 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A131657

For n >= 1, put A_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j and B_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j * (Sum_{k=1..j*n} (1/k)), and let b(n) be the largest integer for which exp(B_n(z)/(b(n)*A_n(z))) has integral coefficients. The sequence is b(n).
(history; published version)

#37 by Peter Luschny at Sun Sep 22 07:25:01 EDT 2024

STATUS

reviewed

approved

#36 by Joerg Arndt at Sun Sep 22 03:20:48 EDT 2024

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proposed

reviewed

#35 by Jason Yuen at Sun Sep 22 03:17:56 EDT 2024

STATUS

editing

proposed

#34 by Jason Yuen at Sun Sep 22 03:17:45 EDT 2024

EXAMPLE

Thus, the conjectured value for a(24) is 24! * (2^{(0-4) * 3^(0-1) * 5^(1-0) * 7^(0-1) * 11^(0-1) * 13^(0-1) * 17^(0-1) * 19^(0-1) * 23^(0-1)) since no exponent of a prime is > 2. This product equals 8691005030400000 = a(24). (End)

STATUS

approved

editing

#33 by Joerg Arndt at Mon May 25 07:53:59 EDT 2020

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reviewed

approved

#32 by Peter Luschny at Mon May 25 04:11:45 EDT 2020

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proposed

reviewed

#31 by Peter Luschny at Mon May 25 03:54:38 EDT 2020

STATUS

editing

proposed

Discussion

Mon May 25

03:57

Michel Marcus: ok understood : you had to tweak the conjecture with Xi(7)=1/140 to make it work, is that it ?  (I did not look at the article)

03:57

Petros Hadjicostas: Thanks. And if anyone figures out their conjecture (in their paper) for A131658, please let me know. They refer to pp. 7-8, but I cannot find it.

04:00

Petros Hadjicostas: Yes, the case n = 7 is an aberration. They explicitly single it out. (The case n = 1 is not because an empty product equals 1.)

#30 by Peter Luschny at Mon May 25 03:54:13 EDT 2020

EXAMPLE

Let us To illustrate the Krattenhaler-Rivoal conjecture for consider the case n = 24. We have Then H_24 = Sum_{k=1..24} 1/k = 1347822955/356948592 and {p <= 24} = {2, 3, 5, 7, 11, 13, 17, 19, 23} with {v_p(numerator): p <= 24} = {0, 0, 1, 0, 0, 0, 0, 0, 0} and {v_p(denominator): p <= 24} = {4, 1, 0, 1, 1, 1, 1, 1, 1}.

Discussion

Mon May 25

03:54

Peter Luschny: Light edits. OK?

#29 by Peter Luschny at Mon May 25 03:53:10 EDT 2020

COMMENTS

According to Krattenhaler and Rivoal (2007-2009), their conjecture is that a(n) = n!*Xi(n), where Xi(1) = 1, Xi(7) = 1/140, and Xi(n) = Product_{p <= n} p^min(2, v_p(H_n)) for n <> 1, 7, where v_p(r) is the p-adic valuation of rational r. (Here p indicates a prime and H_n is the n-th harmonic number.) - Petros Hadjicostas, May 24 2020

STATUS

proposed

editing

#28 by Petros Hadjicostas at Mon May 25 03:47:30 EDT 2020

STATUS

editing

proposed

Discussion

Mon May 25

03:50

Petros Hadjicostas: Yes, this is what I mean: if you do not use the ad hoc value "1/140" and you use n! * Product_{p <= n} p^min(2, v_p(n)) for n=7, you get 108.

03:51

Petros Hadjicostas: 7!*(1/140) = 36