A309829 Numbers k for which A120077(k) != A007407(k).

20, 21, 110, 136, 156, 930, 44310

Offset: 1

Author

Jeppe Stig Nielsen, Aug 18 2019

Keywords

• nonn
• more

Comments

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.

Will gcd(A120077(k), A007407(k)) always be one of the numbers A120077(k) and A007407(k)?

Should this sequence be infinite?

Crossrefs

Cf. A120072, A120077, A007407.

Programs

PARI

s=1; for(n=2, +oo, s += 1/n^2; denominator(s)!=denominator(s-1/n) && print1(n, ", "))