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A290815 Numbers m such that the numerator of Sum_{k=1..m, gcd(k,m) = 1} 1/k is divisible by m^3.

%I A290815 #50 Feb 16 2025 08:33:50
%S A290815 1,39,78,155,310,465,546,793,798,930,1092,1586,1638,1860,2170,2379,
%T A290815 2394,3172,3276,3965,4340,4758,4914,5219,6045,6510,7137,7182,7930,
%U A290815 9516,9828,10374,10438,11102,11895,12090,13020,14274,15657,15860,16843,16891,18135
%N A290815 Numbers m such that the numerator of Sum_{k=1..m, gcd(k,m) = 1} 1/k is divisible by m^3.
%C A290815 A generalization of Wolstenholme primes (A088164) for composite number.
%C A290815 Leudesdorf proved in 1888 that the numerator of Sum_{k=1..n, gcd(k,n)=1} 1/k is divisible by n^2 for all (but not only) numbers n with gcd(n,6)=1, which is a generalization of Wolstenholme's theorem.
%C A290815 Terms that are coprime to 6: 1, 155, 793, 3965, 5219, 16843, 16891, 51305, ...
%C A290815 a(41) = A088164(1) = 16843.
%C A290815 A general conjecture: if, for some e > 0, m^e | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k), then m^(e-1) | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k^2). Note: in this case, the exponent e = 3. Problem: are there numbers m > 1 such that m^4 | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k)? - _Thomas Ordowski_, Aug 10 2019
%C A290815 This general conjecture was checked up to 10^4. This problem has no solution up to 10^5. - _Amiram Eldar_, Aug 10 2019
%C A290815 It appears that all odd terms of this sequence are odd numbers m such that the numerator of Sum_{k=1..m, gcd(k,m)=1} 1/k^2 is divisible by m^2. - _Thomas Ordowski_, Aug 12 2019
%D A290815 G. H. Hardy and E. M. Wright, Introduction to the theory of numbers, 5th edition, Oxford, England: Clarendon Press, 1979, pp. 100-102.
%H A290815 David A. Corneth, <a href="/A290815/b290815.txt">Table of n, a(n) for n = 1..174</a> (terms <= 4*10^5; first 100 terms from Amiram Eldar)
%H A290815 C. Leudesdorf, <a href="https://doi.org/10.1112/plms/s1-20.1.199">Some results in the elementary theory of numbers</a>, Proceedings of the London Mathematical Society, Vol. 20 (1888), pp. 199-212.
%H A290815 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LeudesdorfTheorem.html">Leudesdorf Theorem</a>.
%H A290815 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wolstenholme&#39;s_theorem">Wolstenholme's theorem</a>.
%e A290815 Sum_{k=1..39, gcd(k,39)=1} 1/k = 46855131783993/15222026943200, and 46855131783993 = 39^3 * 789884047, thus 39 is in the sequence.
%t A290815 seqQ[n_] :=  Module[{}, g[m_] := GCD[n, m] == 1; Divisible[Numerator[Plus @@ (1/Select[Range[n], g])], n^3]]; Select[Range[10^5], seqQ]
%o A290815 (PARI) isok(n) = numerator(sum(k=1, n, if (gcd(n, k)==1, 1/k))) % n^3 == 0; \\ _Michel Marcus_, Aug 11 2017
%o A290815 (PARI) upto(n) = {my(v = vector(n), d = divisors(n), res = List(), squarefreepart(n) = factorback(factorint(n)[, 1])); v[1] = 1; for(i = 2, n, v[i] = v[i-1] + 1/i; ); for(j = 1, n, fr = v[j]; d = divisors(squarefreepart(j)); for(i = 2, #d, fr += 1/d[i] * v[j/d[i]] * (-1)^omega(d[i]) ); if(numerator(fr) % j^3 == 0, listput(res, j); ) ); res } \\ _David A. Corneth_, Aug 23 2019
%Y A290815 Cf. A088164, A093600.
%K A290815 nonn
%O A290815 1,2
%A A290815 _Amiram Eldar_, Aug 11 2017