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A143578 A positive integer n is included if j+n/j divides k+n/k for every divisor k of n, where j is the largest divisor of n that is <= sqrt(n).

%I A143578 #16 Sep 01 2019 22:06:11
%S A143578 1,2,3,5,7,11,13,15,17,19,23,29,31,35,37,41,43,47,53,59,61,67,71,73,
%T A143578 79,83,89,95,97,101,103,107,109,113,119,127,131,137,139,143,149,151,
%U A143578 157,163,167,173,179,181,191,193,197,199,209,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,287,293
%N A143578 A positive integer n is included if j+n/j divides k+n/k for every divisor k of n, where j is the largest divisor of n that is <= sqrt(n).
%C A143578 This sequence trivially contains all the primes.
%C A143578 There is no term <= 5*10^7 with bigomega(n)>2, i.e., with more than 2 prime factors. - _M. F. Hasler_, Aug 25 2008. Compare A142591.
%C A143578 If it is always true that the terms have <= 2 prime divisors, then this sequence is equal to {1} U primes U {pq: p, q prime, p+q | p^2-1}. - _David W. Wilson_, Aug 25 2008
%H A143578 Robert Israel, <a href="/A143578/b143578.txt">Table of n, a(n) for n = 1..10000</a>
%e A143578 The divisors of 35 are 1,5,7,35. The sum of the two middle divisors is 5+7 = 12. 12 divides 7 + 35/7 = 5+35/5 = 12, of course. And 12 divides 1 + 35/1 = 35 +35/35 = 36. So 35 is in the sequence.
%p A143578 filter:= proc(n) local k,D,j,t;
%p A143578     if isprime(n) then return true fi;
%p A143578     D:= select(t -> t^2 <= n, numtheory:-divisors(n));
%p A143578   j:= max(D);
%p A143578   t:= j+n/j;
%p A143578   andmap(k -> (k+n/k) mod t = 0, D);
%p A143578 end proc:
%p A143578 select(filter, [$1..1000]); # _Robert Israel_, Sep 01 2019
%t A143578 a = {}; For[n = 1, n < 200, n++, b = Max[Select[Divisors[n], # <= Sqrt[n] &]]; If[ Length[Union[Mod[Divisors[n] + n/Divisors[n], b + n/b]]] == 1, AppendTo[a, n]]]; a (* _Stefan Steinerberger_, Aug 29 2008 *)
%o A143578 (PARI) isA143578(n)={ local( d=divisors(n), j=(1+#d)\2, r=d[ j ]+d[ 1+#d-j ]); for( k=1, j, ( d[k]+d[ #d+1-k] ) % r & return ); 1 }
%o A143578 for(n=1,300,isA143578(n) && print1(n",")) \\ _M. F. Hasler_, Aug 25 2008
%Y A143578 Cf. A063655, A142591.
%K A143578 nonn
%O A143578 1,2
%A A143578 _Leroy Quet_, Aug 24 2008
%E A143578 More terms from _M. F. Hasler_, Aug 25 2008 and _Stefan Steinerberger_, Aug 29 2008