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Numbers m for which max_{2<=k<(m-2)/2} Sum_{d>1: d|m-k, k|m-d} 1 = 3.
10

%I #35 Jan 02 2023 12:30:48

%S 10,15,17,20,23,25,29,31,37,40,41,43,53,67,71,73,79,89,97,109,127,151,

%T 157,181,193,239,241,271,313,331,337,349,373,397,421,433,449,601,613,

%U 661,673,701,757,811,1009,1021,1051,1117,1249,1471,1531,1741

%N Numbers m for which max_{2<=k<(m-2)/2} Sum_{d>1: d|m-k, k|m-d} 1 = 3.

%C All terms a(n) >= 41 are primes. - _Vladimir Shevelev_, May 12 2013

%C If prime p is in the sequence, then either (p-2,p) is a twin prime pair, or p-2 = q*r, where q and r are distinct primes, or p-2 is the cube of a prime. - _Vladimir Shevelev_, May 15 2013

%H Peter J. C. Moses, <a href="/A188579/b188579.txt">Table of n, a(n) for n = 1..215</a>

%H Vladimir Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2013-May/011168.html">Proof that all terms >= 41 of A188579 are primes</a>, SeqFan Mailing List, May 15, 2013.

%e Let n=10. Then k takes the values 2 and 3. If k=3, then d=7 and k divides n-d; if k=2, then d = 2,4,8, n-d = 8,6,2 and k divides all these values. Since max(1,3) = 3, 10 is in the sequence. - _Vladimir Shevelev_, May 12 2013

%Y Cf. A188550.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Apr 04 2011