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

1,2

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.

Links

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

Formula

A351112(a(n)) = 1.

Example

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

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 *)

Crossrefs

Cf. A020492, A351112.

Sequence in context: A007310 A069040 A070191 * A231810 A314294 A363999

Adjacent sequences: A367731 A367732 A367733 * A367735 A367736 A367737

Keyword

nonn

Author

Robert Israel, Nov 28 2023

Status

approved