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%I #20 Sep 08 2022 08:46:21
%S 1,39,42,78,155,156,266,310,465,546,620,793,798,930,1092,1586,1596,
%T 1638,1860,2170,2184,2379,2394,3172,3276,3720,3965,4340,4758,4788,
%U 4914,5219,6045,6344,6510,6552,7137,7182,7930,8680,9516,9828,10374,10438,11102,11895,12090,13020,14274,14364,15657,15860,16843,16891,18135
%N Numbers m such that the numerator of Sum_{k=1..m, gcd(k,m) = 1} 1/k^2 is divisible by m^2.
%C Probably A290815 is a subset of these numbers. How to prove it?
%C Conjecture: odd terms of this sequence are odd terms of A290815.
%C Problem: are there numbers m > 1 such that the numerator of Sum_{k=1..m, gcd(k,m)=1} 1/k^2 is divisible by m^3 ?
%t aQ[n_] := Divisible[Numerator[Plus @@ ((1/Select[Range[n], CoprimeQ[n, #] &])^2)], n^2]; Select[Range[10^4], aQ]
%o (PARI) isok(m) = !(numerator(sum(k=1, m, if (gcd(k,m) == 1, 1/k^2))) % m^2); \\ _Michel Marcus_, Aug 13 2019
%o (Magma) v:= [Numerator(&+[1/k^2:k in [1..n]|Gcd(k,n) eq 1]):n in [1..10000]]; [m:m in [1..#v]| v[m] mod m^2 eq 0]; // _Marius A. Burtea_, Aug 14 2019
%Y Cf. A290815.
%K nonn
%O 1,2
%A _Amiram Eldar_ and _Thomas Ordowski_, Aug 13 2019
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