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

2, 3, 4, 5, 7, 8, 10, 14, 36, 37, 39, 42, 43, 44, 46, 47

Offset: 1

Author

Paolo P. Lava, Mar 06 2013

Keywords

• nonn
• fini
• full

Comments

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

Examples

			2 is a term because floor(sqrt(1))+floor(sqrt(2)) = 1+1 = 2 is prime;
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.
		

Crossrefs

Primes in A022554.

Cf. A220953.

Programs

Maple

A214036:=proc(q)  local a,n; a:=0;
for n from 1 to q do a:=a+floor(sqrt(n)); if isprime(a) then print(n); fi; od; end:
A214036(10^10);
Alternative program:
A214036_bis:=proc(q)  local a,j,n; a:=0;
for n from 1 to q do for j from 1 to 2*n+1 do
    a:=a+n; if isprime(a) then print(n^2+j-1); fi;
od; od; end:
A214036_bis(10^10);

Mathematica

Position[Accumulate[Table[Floor[Sqrt[n]],{n,50}]],?PrimeQ]//Flatten (* _Harvey P. Dale, Apr 14 2017 *)

PARI

sm = 0; for (n=1, 10^9, sm+=sqrtint(n); if (isprime(sm), print1(n,", ")));
/* Joerg Arndt, Mar 07 2013 */