OFFSET
7,2
COMMENTS
Conjecture: If x is a whole number greater than 1, the Sum_{k>=1} (k^(1/k^(1 + 1/sqrt(x))) - 1) = x + C, where C is a constant less than 1.
The above relation was tested for all 1 < x < 10^7.
If x = 1, the sum is A329117.
This sum demonstrates this relationship: setting sqrt(x) = 1111 generates the sum 1111^2 + C or 1234321 + C. Another example would be Sum_{k>=1} (k^(1/k^(1 + 1/sqrt(1729))) - 1) = 1729.84841430674....
Evaluating the sum at larger x values converges slower and slower. Monotonically changing extrapolation methods such as Richardson's Extrapolation must be used to compute these values.
Since the output (x + C) will be the square of the input (sqrt(x)) plus a constant less than 1, this implies that Sum_{k>=1} (k^(1/k^(1 + 1/sqrt(x))) - 1) diverges as x tends to infinity, or simplified to Sum_{k>=1} (k^(1/k) - 1).
EXAMPLE
1234321.98281389093336864244004887748682691258771548...
MATHEMATICA
digits = 120; d = 1; j = 1; s = 0; While[Abs[d] > 10^(-digits - 5), d = (-1)^j/j!*Derivative[j][Zeta][(1 + 1/1111)*j]; s += d; j++]; RealDigits[s, 10, 120][[1]] (* Vaclav Kotesovec, Jun 18 2023 *)
PROG
(PARI) sumpos(k=1, k^(1/(k^(1 + 1/1111))) - 1).
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Daniel Hoyt, Jan 19 2022
STATUS
approved