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

1,1

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Intersection of A003601 and A140826.

Sequence in context: A171014 A254050 A250094 * A285516 A319801 A339907

Adjacent sequences: A351395 A351396 A351397 * A351399 A351400 A351401

Keyword

nonn

Author

Ctibor O. Zizka, Feb 09 2022

Status

approved