A142591 -id:A142591 - cp's OEIS frontend

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

Leroy Quet, Aug 24 2008

nonn

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

			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.

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

Cf. A063655, A142591.

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

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 *)

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

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

A238232 Composite numbers n such that the sum of numbers x<=n not coprime to n divides the sum of numbers y<=n coprime to n.

Original entry on oeis.org

15, 35, 95, 119, 143, 209, 255, 287, 319, 323, 377, 527, 559, 779, 899, 923, 989, 1007, 1189, 1199, 1295, 1343, 1349, 1763, 1919, 2159, 2507, 2759, 2911, 3239, 3599, 3827, 4031, 4607, 5183, 5207, 5249, 5459, 5543, 6439, 6479, 6887, 7067, 7279, 7739, 8159, 8639

Offset: 1

Paolo P. Lava, Feb 21 2014

nonn

Also numbers n such that n+1-phi(n) | phi(n).

A203966 lists the numbers n such that the sum of numbers x<=n coprime to n divides the sum of numbers y<=n not coprime to n. This is equivalent to numbers n such that phi(n) | n+1. [suggested by Giovanni Resta]

			The numbers coprime to 15 are 1, 2, 4, 7, 8, 11, 13, 14 and their sum is 60. In fact 15*phi(15)/2 = 60.
The sum of the numbers from 1 to 15 is 15*(15+1)/2 = 120 and therefore the sum of the numbers not coprime to 15 is 120 - 60 = 60. At the end we have that 60/60 = 1.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A061367, A050474, A070161, A142591, A203966.

with(numtheory);P:=proc(q) local i,n;
for n from 2 to q do if not isprime(n) then
if type(phi(n)/(n+1-phi(n)),integer) then print(n);
fi; fi; od; end: P(10^6);

cp's OEIS Frontend

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).

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