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Numbers that have no balanced divisors except for 1.
1

%I #16 Dec 01 2023 15:55:43

%S 1,5,7,11,13,17,19,23,25,29,31,37,41,43,47,49,53,55,59,61,65,67,71,73,

%T 77,79,83,85,89,91,95,97,101,103,107,109,113,115,119,121,125,127,131,

%U 133,137,139,143,145,149,151,155,157,161,163,167,169,173,179,181,185,187,191,193,197,199,203

%N Numbers that have no balanced divisors except for 1.

%C Numbers k such that A351112(k) = 1.

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

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

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

%H Robert Israel, <a href="/A367734/b367734.txt">Table of n, a(n) for n = 1..10000</a>

%F A351112(a(n)) = 1.

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

%p filter:= proc(n) uses numtheory;

%p andmap(t -> sigma(t) mod phi(t) <> 0, divisors(n) minus {1})

%p end proc:

%p select(filter, [$1..1000]);

%t Select[Range[200], DivisorSum[#, 1 &, Divisible[DivisorSigma[1, #1], EulerPhi[#1]] &] == 1 &] (* _Amiram Eldar_, Nov 28 2023 *)

%Y Cf. A020492, A351112.

%K nonn

%O 1,2

%A _Robert Israel_, Nov 28 2023