A143578 - OEIS

%I #16 Sep 01 2019 22:06:11

%S 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 79,83,89,95,97,101,103,107,109,113,119,127,131,137,139,143,149,151,

%U 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 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 This sequence trivially contains all the primes.

%C 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 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 Robert Israel, <a href="/A143578/b143578.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 filter:= proc(n) local k,D,j,t;

%p     if isprime(n) then return true fi;

%p     D:= select(t -> t^2 <= n, numtheory:-divisors(n));

%p   j:= max(D);

%p   t:= j+n/j;

%p   andmap(k -> (k+n/k) mod t = 0, D);

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Sep 01 2019

%t 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 (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 for(n=1,300,isA143578(n) && print1(n",")) \\ _M. F. Hasler_, Aug 25 2008

%Y Cf. A063655, A142591.

%K nonn

%O 1,2

%A _Leroy Quet_, Aug 24 2008

%E More terms from _M. F. Hasler_, Aug 25 2008 and _Stefan Steinerberger_, Aug 29 2008