A351398 Numbers k >= 3 such that the arithmetic mean of the divisors of k AND the arithmetic mean of the nondivisors of k are integers.

3, 5, 7, 11, 13, 17, 19, 20, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Author

Ctibor O. Zizka, Feb 09 2022

Keywords

• nonn

Comments

This sequence includes all primes >= 3 because A000203(p)/A000005(p) = (p + 1)/2 AND (A000217(p) - A000203(p))/A049820(p) = (p + 1)/2.

Up to 2 * 10^8 the only nonprime terms are 20 and 432. - Robert Israel, May 06 2024

Examples

			k = 13, A000203(13)/A000005(13) = 14/2 = 7, A024816(13)/A049820(13) = 77/11 = 7, so 13 is a term.
k = 20, A000203(20)/A000005(20) = 42/6 = 7, A024816(20)/A049820(20) = 168/14 = 12, so 20 is a term.
		

Links

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

Crossrefs

Cf. A000005, A000203, A000217, A024816, A049820.

Intersection of A003601 and A140826.

Programs

Maple

filter:= proc(n) local s,t;
  s:= numtheory:-sigma(n);
  t:= numtheory:-tau(n);
  (s/t)::integer and ((n*(n+1)/2 - s)/(n-t))::integer;
end proc:
select(filter, [$2..1000]); # Robert Israel, May 06 2024

Mathematica

Select[Range[2, 350], Divisible[(s = DivisorSigma[1, #]), (d = DivisorSigma[0, #])] && Divisible[#*(# + 1)/2 - s, # - d] &] (* Amiram Eldar, Feb 09 2022 *)

PARI

isok(k) = if (k>=3, my(sk=sigma(k), nk=numdiv(k), tk=k*(k+1)/2); !(sk % nk) && !((tk - sk) % (k - nk))); \\ Michel Marcus, Feb 10 2022