[seqfan] Discussion of A14960.

N. J. A. Sloane njas at research.att.com
Sat May 22 18:52:07 CEST 2010

The present entry is:

%I A014960
%S A014960 1,23,529,1081,12167,24863,50807,279841,571849,1168561,2387929,2870377
%N A014960 Numbers n such that n divides s(n), where s(1)=1, s(k)=s(k-1)+k*24^(k-1) (A014942).
%C A014960 Initial terms are 23^n, 23^(n-1)*47, 23^(n-2)*47^2,...23*47^(n-1),23^(n+1), etc. with som\
etime a little "noise" between terms (eg.: for a(12)=23*124799 between a(11)=23*47^3 and maybe a(13)\
=23^5). Maybe another sequence is interlaced, which would involve 23^n, 23^(n-1)*124799, etc., in wh\
ich case an infinity of products of powers of 23 and powers of another prime factor may occur in the\
 sequence. Conjecture: Next term, a(13), very probably is 23^5. Conjecture: All numbers in the seque\
nce are multiple of 23. Conjecture: All numbers in the sequence have at most two different prime fac\
tors. - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Oct 10 2003
%t A014960 s = 1; Do[ If[ Mod[ s, n ] == 0, Print[n]]; s = s + (n + 1)*24^n, {n, 1, 100000}]
%Y A014960 Cf. A014942.
%Y A014960 Sequence in context: A171328 A097778 A057193 this_sequence A171297 A009967 A147642
%Y A014960 Adjacent sequences: A014957 A014958 A014959 this_sequence A014961 A014962 A014963
%K A014960 nonn
%O A014960 1,2
%A A014960 Olivier Gerard (olivier.gerard(AT)gmail.com)
%E A014960 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 13 2000
%E A014960 Four more terms from Thomas Baruchel (baruchel(AT)users.sourceforge.net), Oct 10 2003

Alex Adamchuk has proposed some major edits to this, as follows:

%S A014960 1,23,529,1081,12167,24863,50807,279841,571849,1168561,2387929,2870377,
%T A014960 6436343,7009273,13152527,15954479,26876903,54922367,66018671,112232663,
%U A014960 134907719,148035889,161213279,302508121,329435831,366953017,537539141
%E A014960 a(13)-a(32) from Alexander Adamchuk (alex(AT)kolmogorov.com), May 16 2010
%C A014960 Contribution from Alexander Adamchuk (alex(AT)kolmogorov.com), May 16 2010: (Start)
%C A014960 Better definition: Numbers n such that n divides 24^n - 1.
%C A014960 First two contrexamples for conjecture by Thomas Baruchel "...at most two different prime factors": 15954479 = 23*47*14759, 134907719 = 23*47*124799.
%C A014960 Prime factors of a (n) in the order of their appearance are {23, 47, 124799, 304751, 14759, 497261, 49727, ...} = A087807. (End)
%Y A014960 Contribution from Alexander Adamchuk (alex(AT)kolmogorov.com), May 16 2010: (Start)
%Y A014960 Cf. A087807 = Prime factors of terms occurring in A014960.
%Y A014960 Cf. A128356 = Least number k>1 (that is not the power of prime p) such that k divides (p+1)^k-1, where p = Prime[n]. (End)

I don't have time to study this.  Are these changes correct?   I like to see proofs
before I make such drastic changes.

Thanks,  Neil

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