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%I #9 Mar 10 2023 10:45:38
%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 38,39,41,42,43,44,45,46,47,48,49,51,53,54,55,56,57,59,60,61,62,63,65,
%U 66,67,69,70,71,73,75,76,77,78,79,81,83,84,85,86,87,89,91
%N Infinitary arithmetic numbers: numbers for which the arithmetic mean of the infinitary divisors is an integer.
%C Number k such that A037445(k) divides A049417(k).
%C Subsequence of the unitary arithmetic numbers (A103826).
%H Amiram Eldar, <a href="/A361386/b361386.txt">Table of n, a(n) for n = 1..10000</a>
%F 6 is a term since the arithmetic mean of its infinitary divisors, {1, 2, 3, 6}, is 3 which is an integer.
%t f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, (1 + p^(2^(m - j)))/2, 1], {j, 1, m}]]; q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], q]
%o (PARI) is(n) = {my(f = factor(n), b); denominator(prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], (f[i, 1]^(2^(#b-k))+1)/2, 1)))) == 1; }
%Y Cf. A037445, A049417, A077609.
%Y Similar sequences: A003601, A103826.
%K nonn
%O 1,2
%A _Amiram Eldar_, Mar 10 2023
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