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Bi-unitary arithmetic numbers: numbers for which the arithmetic mean of the bi-unitary divisors is an integer.
2

%I #9 Mar 24 2023 11:12:55

%S 1,3,5,6,7,9,11,12,13,14,15,17,19,21,22,23,25,27,28,29,30,31,33,35,37,

%T 38,39,41,42,43,44,45,46,47,49,51,53,54,55,56,57,59,60,61,62,63,65,66,

%U 67,69,70,71,73,75,76,77,78,79,81,83,84,85,86,87,89,91,92,93,94,95,96,97,99

%N Bi-unitary arithmetic numbers: numbers for which the arithmetic mean of the bi-unitary divisors is an integer.

%C First differs from A361386 at n = 35.

%C Number k such that A286324(k) divides A188999(k).

%H Amiram Eldar, <a href="/A361786/b361786.txt">Table of n, a(n) for n = 1..10000</a>

%e 6 is a term since the arithmetic mean of its bi-unitary divisors, {1, 2, 3, 6}, is 3 which is an integer.

%t f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e]; q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], q]

%o (PARI) is(n) = {my(f = factor(n), p, e); denominator(prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e))) == 1; }

%Y Cf. A188999, A222266, A286324.

%Y Similar sequences: A003601, A103826, A361386.

%K nonn

%O 1,2

%A _Amiram Eldar_, Mar 24 2023