cp's OEIS Frontend

A214036 Numbers k such that floor(sqrt(1)) + floor(sqrt(2)) + floor(sqrt(3)) + ... + floor(sqrt(k)) is prime.

%I A214036 #25 Feb 19 2025 01:02:59
%S A214036 2,3,4,5,7,8,10,14,36,37,39,42,43,44,46,47
%N A214036 Numbers k such that floor(sqrt(1)) + floor(sqrt(2)) + floor(sqrt(3)) + ... + floor(sqrt(k)) is prime.
%C A214036 The sequence is complete. Indeed, let s(n) be the sum of floor(sqrt(k)) for k from 1 to n. It is easy to verify that s(n^2+j), for 0 <= j < (n+1)^2-n^2, is equal to n(j+1) + n(4n+1)(n-1)/6, which is always divisible by n or by n/6 for n > 6. - _Giovanni Resta_, Mar 26 2014
%e A214036 2 is a term because floor(sqrt(1))+floor(sqrt(2)) = 1+1 = 2 is prime;
%e A214036 14 is a term because floor(sqrt(1))+ ... +floor(sqrt(14)) = 1+1+1+2+2+2+2+2+3+3+3+3+3+3 = 31 is prime.
%p A214036 A214036:=proc(q)  local a,n; a:=0;
%p A214036 for n from 1 to q do a:=a+floor(sqrt(n)); if isprime(a) then print(n); fi; od; end:
%p A214036 A214036(10^10);
%p A214036 Alternative program:
%p A214036 A214036_bis:=proc(q)  local a,j,n; a:=0;
%p A214036 for n from 1 to q do for j from 1 to 2*n+1 do
%p A214036     a:=a+n; if isprime(a) then print(n^2+j-1); fi;
%p A214036 od; od; end:
%p A214036 A214036_bis(10^10);
%t A214036 Position[Accumulate[Table[Floor[Sqrt[n]],{n,50}]],_?PrimeQ]//Flatten (* _Harvey P. Dale_, Apr 14 2017 *)
%o A214036 (PARI)
%o A214036 sm = 0; for (n=1, 10^9, sm+=sqrtint(n); if (isprime(sm), print1(n,", ")));
%o A214036 /* _Joerg Arndt_, Mar 07 2013 */
%Y A214036 Primes in A022554.
%Y A214036 Cf. A220953.
%K A214036 nonn,fini,full
%O A214036 1,1
%A A214036 _Paolo P. Lava_, Mar 06 2013