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A327054 a(n) is the smallest number m such that the antiharmonic mean of the divisors is n, or 0 if no such m exists.

%I A327054 #24 Sep 20 2024 11:06:18
%S A327054 1,0,4,0,0,0,9,0,0,0,16,0,20,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,
%T A327054 0,50,0,0,0,0,0,0,0,49,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,81,0,100,0,0,
%U A327054 0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0
%N A327054 a(n) is the smallest number m such that the antiharmonic mean of the divisors is n, or 0 if no such m exists.
%C A327054 a(n) = the smallest number m such that sigma_2(n) / sigma_1(n) = A001157(m) / A000203(m) = n, or 0 if no such m exists.
%C A327054 Zeros occur if n is not in A176799.
%C A327054 See A000290, A091911 and A162538 for like sequences for geometric, arithmetic and harmonic means of the divisors.
%H A327054 Robert Israel, <a href="/A327054/b327054.txt">Table of n, a(n) for n = 1..10000</a>
%e A327054 a(3) = 4 because 4 is the smallest number m with sigma_2(m) / sigma_1(m) = 3; sigma_2(4) / sigma_1(4) = 21 / 7 = 3.
%p A327054 # This uses the b-file for A004394
%p A327054 # See comment at A176799
%p A327054 K:= 100: # to get terms <= K
%p A327054 M:= 36 * K^2/Pi^4:
%p A327054 for i from 1 while A004394[i] < M do od:
%p A327054 r:= numtheory:-sigma(A004394[i])/A004394[i]:
%p A327054 V:= Vector(K):
%p A327054 for m from 1 to r*K do
%p A327054   F:= numtheory:-divisors(m);
%p A327054   v:= add(d^2, d=F)/add(d, d=F);
%p A327054   if v::integer and v <= K and V[v] = 0 then V[v]:= m fi;
%p A327054 od:
%p A327054 convert(V,list); # _Robert Israel_, Sep 05 2024
%o A327054 (Magma) A327054:=func<n|exists(r){m:m in[1..10000] |  IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m)) and (&+[d^2: d in Divisors(m)] / SumOfDivisors(m)) eq n}select r else 0>; [A327054(n): n in[1..100]];
%Y A327054 Cf. A020487, A176797, A176799, A176800.
%Y A327054 Cf. A000290, A004394, A091911, A162538.
%K A327054 nonn
%O A327054 1,3
%A A327054 _Jaroslav Krizek_, Oct 06 2019