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A328944 Arithmetic numbers (A003601) that are not harmonic (A001599).

%I A328944 #22 May 05 2025 08:08:05
%S A328944 3,5,7,11,13,14,15,17,19,20,21,22,23,27,29,30,31,33,35,37,38,39,41,42,
%T A328944 43,44,45,46,47,49,51,53,54,55,56,57,59,60,61,62,65,66,67,68,69,70,71,
%U A328944 73,77,78,79,83,85,86,87,89,91,92,93,94,95,96,97,99,101
%N A328944 Arithmetic numbers (A003601) that are not harmonic (A001599).
%C A328944 Numbers m such that the arithmetic mean of the divisors of m is an integer but the harmonic mean of the divisors of m is not an integer.
%C A328944 Numbers m such that A(m) = A000203(m)/A000005(m) is an integer but H(m) = m * A000005(m)/A000203(m) is not an integer.
%C A328944 Corresponding values of A(m): 2, 3, 4, 6, 7, 6, 6, 9, 10, 7, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...
%C A328944 Corresponding values of H(m): 3/2, 5/3, 7/4, 11/6, 13/7, 7/3, 5/2, 17/9, 19/10, 20/7, 21/8, 22/9, ...
%C A328944 Complement of A007340 with respect to A003601.
%H A328944 Robert Israel, <a href="/A328944/b328944.txt">Table of n, a(n) for n = 1..10000</a>
%p A328944 harm:= proc(S) local s; nops(S)/add(1/s,s=S) end proc:
%p A328944 filter:= proc(n) local S;
%p A328944   S:= numtheory:-divisors(n);
%p A328944   (convert(S,`+`)/nops(S))::integer and not harm(S)::integer
%p A328944 end proc:
%p A328944 select(filter, [$1..200]); # _Robert Israel_, May 04 2025
%t A328944 Select[Range[100], Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] && !Divisible[# * DivisorSigma[0, #], DivisorSigma[1, #]] &] (* _Amiram Eldar_, Nov 01 2019 *)
%o A328944 (Magma) [m: m in [1..10^5] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m))];
%Y A328944 Cf. A000005, A000203, A003601, A007340, A046999, A328945.
%K A328944 nonn
%O A328944 1,1
%A A328944 _Jaroslav Krizek_, Oct 31 2019