A367734 Numbers that have no balanced divisors except for 1.

1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203

Offset: 1

Author

Robert Israel, Nov 28 2023

Keywords

• nonn

Comments

Numbers k such that A351112(k) = 1.

Includes all primes except for 2 and 3, and all powers of those primes.

If k is a term, then so are all divisors of k.

For i < 271, a(i+68) = a(i) + 210, and this equation seems to be true for most i.

Examples

			a(9) = 25 is a term because of its divisors 1, 5, 25, only 1 is balanced.
		

Links

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

Crossrefs

Cf. A020492, A351112.

Programs

Maple

filter:= proc(n) uses numtheory;
  andmap(t -> sigma(t) mod phi(t) <> 0, divisors(n) minus {1})
end proc:
select(filter, [$1..1000]);

Mathematica

Select[Range[200], DivisorSum[#, 1 &, Divisible[DivisorSigma[1, #1], EulerPhi[#1]] &] == 1 &] (* Amiram Eldar, Nov 28 2023 *)

Formula

A351112(a(n)) = 1.