a discrepancy concerning A063979

Tue Jul 8 14:09:30 CEST 2008

I confirm the current values of A063979.
Apparently, the default real precision in Python is not enough to get
a correct answer for n>=17.
I suggest Dmitry to increase precision and recompute the sequence with
the same program.

Regards,
Max

On Tue, Jul 8, 2008 at 4:37 AM, N. J. A. Sloane <njas at research.att.com> wrote:
> Dear Sequence Fans, At present the entry reads as follows:
>
> %I A063979
> %S A063979 1,1,1,3,24,199,1747,16474,168187,1859934,22228104,286078171,
> %T A063979 3949867548,58284826485,915905054360,15276520209206,269617872744249,
> %U A063979 5021159048900643,98417586560408168,2025488254833817394
> %N A063979 Number of decimal digits in (n!)!; A000197.
> %H A063979 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Factorial.html">Factorial</a>
> %p A063979 seq(length((n)!!), n=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 10 2007
> %t A063979 Do[ Print[ Floor[ N[ Log[10, (n!)! ], 24] + 1]], {n, 0, 11} ]
> %Y A063979 Cf. A000197, A063944, A034886.
> %K A063979 base,nonn
> %O A063979 0,4
> %A A063979 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 05 2001
> %E A063979 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 06 2001
>
> Dmitry Kamenetsky (dkamen at rsise.anu.edu.au), Jul 08 2008 says:
>
> %C A063979 for n=17, 18 and 19 I get 5021159048900644, 98417586560408160 and 20254882548338170
> 88 respectively. This is slightly different to the quoted numbers. Can someone please verify t
> his?
>
> %o A063979 Here is my Python program:
>
> from math import sqrt, pi, log, e, floor
>
> def fact(n):
>       r = 1
>       for i in xrange(2, n+1):
>               r = r * i
>       return r
>
>
> def fact2(n):
>
>        n=fact(n)
>        b=(log(2*pi)+log(n))/2+(log(n)-1)*n
>        digits = int(floor(b/log(10)) + 1)
>        return digits
>
>
> for n in xrange(1,20):
>        print n,fact2(n)
>
> (end of quote)
>
> Which version is correct?
>
> Thanks
> Neil
>
>

Hello Andrew,

does some numbers n exist, where the following is true :

n,  RMS(n),  RMS(RMS(n)),  RMS(RMS(RMS(n))), .......  , ends in a cycle or fixed point. (1)
RMS(n) denotes root mean square of divisors of n, RMS(n) is an integer. 

This proposal may not hold strongly for a set of divisors of n,
where RMS(A_1, A_2, ......, A_k) is an integer and (1) holds in this weaker formulation.

Ctibor

> ------------ Pùvodní zpráva ------------
> Od: Andrew Weimholt <andrew at weimholt.com>
> Pøedmìt: More sequences related to RMS Numbers (A140480)
> Datum: 08.7.2008 00:53:03
> ----------------------------------------
> I've submitted the following 5 sequences related to A140480.
> 
> A141812: RMS Values of the RMS Numbers (A140480)
> A141813: Primitive RMS Numbers (not a product of smaller RMS Numbers)
> A141814: RMS Values of Primitive RMS Numbers (A141813)
> A141815: RMS Numbers with non-unique RMS-Values
> A141816: RMS Values for A141815
> 
> The b-files for the first three are attached.
> I didn't create b-files for the last two, because for A141815, if you
> try to include too many terms from the list of known terms of A140480,
> you run the risk of erroneously excluding terms. For example, a term
> in A140480 may appear to have a unique RMS Value if you only look at
> the RMS Values of the first X terms of A140480, but may in fact share
> an RMS Value with A140480(X+k).
> 
> %I A141812
> %S A141812 1, 5, 29, 169, 145, 845, 1105, 2405, 3445, 4901, 2665,
> 5525, 9425, 12325, 12025,
> 17225, 24505, 13325, 32045, 55205, 47125, 61625, 69745, 101065, 99905,
> 77285, 124501,
> 160225, 186745, 204425, 239425, 160225, 273325, 276025, 292825,
> 226525, 446165, 456025,
> 357425, 406445, 456025, 348725, 801125, 582205, 450385, 637325,
> 493025, 505325, 499525
> %N A141812 RMS Values of the RMS Numbers: a(n) is the Root Mean Square
> of the divisors of A140480(n)
> %e A141812 a(5)=145, because A140480(5)=287, with divisors 1,7,41,287
> and RMS(1,7,41,287) = 145.
> %Y A141812 Cf. A140480, A141813, A141814, A141815, A141816.
> %O A141812 1
> %K A141812 ,nonn,
> %A A141812 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008
> 
> 
> %I A141813
> %S A141813 1, 7, 41, 239, 3055, 6665, 9545, 9855, 26095, 34697,
> 155287, 380511, 421655, 627215,
> 814463, 823537, 1166399, 1204281, 1256489, 1289441, 1815073, 2265353,
> 2544697, 2627343,
> 3132935, 3188809, 3762639, 4647985, 4730879, 4963127, 4995569,
> 5054015, 5143945, 6542705,
> 6956927, 9369319, 9963071, 10286649, 10359689, 10707903, 11369969,
> 11538505, 12158135
> %N A141813 Primitive RMS Numbers: RMS Numbers which are not the
> product of two smaller RMS Numbers.
> %C A141813 RMS Numbers (see A140480) are numbers such that the RMS
> (Root Mean Square) of their divisors is an integer. If A and B both
> appear in A140480 and GCD(A,B)=1, then A*B is also in A140480. This
> sequence contains only those RMS numbers that
> are not a product smaller RMS numbers.
> %e A141813 The RMS Number 287 is not in the sequence because 287=7*41
> and both 7 and 41 are RMS Numbers.
> %Y A141813 Cf. A140480, A141812, A141814, A141815, A141816.
> %O A141813 1
> %K A141813 ,nonn,
> %A A141813 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008
> 
> 
> %I A141814
> %S A141814 1, 5, 29, 169, 1105, 2405, 3445, 2665, 9425, 12325, 55205,
> 101065, 124501, 160225, 204425,
> 239425, 292825, 226525, 446165, 456025, 456025, 801125, 637325,
> 493025, 801125, 801125,
> 706225, 1185665, 1185665, 1759925, 1770305, 1291225, 1313845, 1185665,
> 1743625, 6625109,
> 2497625, 1932125, 3663925, 2010025, 4032145, 2953925, 3112525,
> 4032145, 4254445, 6326125
> %N A141814 RMS Values of the Primitive RMS Numbers: a(n) is the Root
> Mean Square of the divisors of A141813(n)
> %e A141814 a(5)=1105, because A141813(5)=3305, with divisors
> 1,5,13,47,65,235,611,3055
> and RMS(1,5,13,47,65,235,611,3055) = 1105.
> %Y A141814 Cf. A140480, A141812, A141813, A141815, A141816.
> %O A141814 1
> %K A141814 ,nonn,
> %A A141814 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008
> 
> 
> %I A141815
> %S A141815 627215, 876785, 1289441, 1815073, 2265353, 3132935,
> 3188809, 4390505, 4647985, 4730879, 6542705, 9026087, 11369969,
> 12705511, 15203889, 15857471, 17888153, 18253913, 18578719, 20871649,
> 21026655, 21930545, 22321663, 23630711,
> 24738935, 27857871, 29160361, 32535895, 32810935, 33116153, 33392983,
> 34357905, 35378105, 36007615, 38598255,
> 45761033, 45798935, 49375521, 52867081, 53926695, 60447311, 62336569,
> 63286535, 74417993, 79589783, 83597345,
> 86369945, 92879473, 104475337, 106427223
> %N A141815 RMS Numbers with non-unique RMS Values
> %e A141815 627215 is an RMS Number with an RMS value of 160225.
> 876785 is an RMS Number with an RMS value of 160225.
> Since these two RMS Numbers have the same RMS Value, their RMS Values are
> non-unique, and therefore they belong to the sequence.
> %Y A141815 Cf. A140480, A141812, A141813, A141814, A141816.
> %O A141815 1
> %K A141815 ,nonn,
> %A A141815 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008
> 
> 
> 
> %I A141816
> %S A141816 160225, 160225, 456025, 456025, 801125, 801125, 801125,
> 801125, 1185665, 1185665, 1185665, 2280125, 4032145,
> 2280125, 4032145, 4005625, 6326125, 6456125, 6569225, 5226065,
> 4032145, 4005625, 4005625, 5928325, 6326125,
> 5226065, 10310885, 5928325, 5928325, 5928325, 5928325, 6569225,
> 6456125, 6569225, 5226065, 16182725, 5928325,
> 6569225, 13224725, 10310885, 21376225, 22079005, 16182725, 13224725,
> 20160725, 21376225, 22079005, 23232625,
> 26130325, 20160725
> %N A141816 RMS Values of the RMS Numbers with non-unique RMS Values:
> a(n) is the Root Mean Square of the divisors of A141815(n).
> %e A141816 a(1)=160225, because A141815(1)=627215, and the Root Mean
> Square of the divisors of 627215 is 160225.
> %Y A141816 Cf. A140480, A141812, A141813, A141814, A141815.
> %O A141816 1
> %K A141816 ,nonn,
> %A A141816 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008
> 
> Andrew
> 
> 
> 

Thanks to everyone who responded!  I will add a note saying
that the present entries are correct.

Neil