OFFSET
1,2
REFERENCES
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd Edition, Addison-Wesley, 1994, Eq. 3.27 on page 87.
D. E. Knuth, The Art of Computer Programming, Vol. 1, 3rd Edition, Addison-Wesley, 1997, Ex. 43 of section 1.2.4.
LINKS
M. Griffiths, More Sums Involving the Floor Function, Math. Gaz., 86 (2002), 285-287.
Maths StackExchange, Find a formula for Sum_{k=1..n} floor(sqrt(k)), see achille hui formula.
Wikipedia, Faulhaber's formula
FORMULA
T(n,k) = Sum_{j=0..k} floor(j^(1/n)).
T(n,k) = (1+k)*floor(k^(1/n)) - (1/(n+1))*Sum_{j=1..n+1} (1 + floor(k^(1/n)))^j*binomial(n+1, j)*Bernoulli(n+1-j).
T(n,k) = (1+k)*floor(k^(1/n)) - Sum_{j=1..floor(k^(1/n))} j^n. - Daniel Hoying, Jun 11 2020
EXAMPLE
The fifth entry in the second row of this array is 7, since 7 = floor(sqrt(1)) + floor(sqrt(2)) + floor(sqrt(3)) + floor(sqrt(4)) + floor(sqrt(5)).
The table array begins:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
1, 2, 3, 5, 7, 9, 11, 13, 16, 19, ...
1, 2, 3, 4, 5, 6, 7, 9, 11, 13, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
...
MATHEMATICA
T[n_, k_] := (1 + k) Floor[k^(1/n)] - HarmonicNumber[Floor[k^(1/n)], -n] (* Daniel Hoying, Jun 11 2020 *)
PROG
(PARI) T(n, k) = sum(j=0, k, sqrtnint(j, n)); \\ Michel Marcus, Mar 12 2016
CROSSREFS
KEYWORD
AUTHOR
John M. Campbell, Mar 11 2016
STATUS
approved