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).

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 101, 103, 107, 109, 113, 119, 127, 131, 137, 139, 143, 149, 151, 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

Offset: 1

Author

Leroy Quet, Aug 24 2008

Keywords

• nonn

Comments

This sequence trivially contains all the primes.

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.

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A063655, A142591.

Programs

Maple

filter:= proc(n) local k,D,j,t;
    if isprime(n) then return true fi;
    D:= select(t -> t^2 <= n, numtheory:-divisors(n));
  j:= max(D);
  t:= j+n/j;
  andmap(k -> (k+n/k) mod t = 0, D);
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 01 2019

Mathematica

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 *)

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 }
for(n=1,300,isA143578(n) && print1(n",")) \\ M. F. Hasler, Aug 25 2008

Extensions

More terms from M. F. Hasler, Aug 25 2008 and Stefan Steinerberger, Aug 29 2008