This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309829 #15 Aug 20 2019 21:46:10
%S A309829 20,21,110,136,156,930,44310
%N A309829 Numbers k for which A120077(k) != A007407(k).
%C A309829 The sequence of rationals related to A120077 is f(k) = Sum_{j=1..k-1} (1/j^2 - 1/k^2), motivated by each term's interpretation as the energy difference between shells k and j in a hydrogen atom model. This can easily be seen to be equal to f(k) = (Sum_{j=1..k} 1/j^2) - 1/k. Compare this with g(k) = Sum_{j=1..k} 1/j^2 which is the starting point for A007407. The question is, when does the final subtraction of 1/k change the denominator (in lowest term)? In one case (k=21), the denominator belonging to f(k) is greater than that belonging to g(k). In cases k=20, 110, 136, 156, 930, 44310, the opposite is true.
%C A309829 Will gcd(A120077(k), A007407(k)) always be one of the numbers A120077(k) and A007407(k)?
%C A309829 Should this sequence be infinite?
%o A309829 (PARI) s=1; for(n=2, +oo, s += 1/n^2; denominator(s)!=denominator(s-1/n) && print1(n, ", "))
%Y A309829 Cf. A120072, A120077, A007407.
%K A309829 nonn,more
%O A309829 1,1
%A A309829 _Jeppe Stig Nielsen_, Aug 18 2019