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A214036 Numbers n such that floor(sqrt(1)) + floor(sqrt(2)) + floor(sqrt(3)) + ... + floor(sqrt(n)) is prime. 1
2, 3, 4, 5, 7, 8, 10, 14, 36, 37, 39, 42, 43, 44, 46, 47 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..16.

EXAMPLE

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.

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 *)

PROG

(PARI)

default(realprecision, 66);

sm = 0; /* sum(n>=1, floor(sqrt(n)) */

for (n=1, 10^9, sm+=sqrtint(n); if (isprime(sm), print1(n, ", ")));

/* Joerg Arndt, Mar 07 2013 */

CROSSREFS

Cf. A220953.

Sequence in context: A339279 A034296 A075745 * A100289 A255130 A054021

Adjacent sequences:  A214033 A214034 A214035 * A214037 A214038 A214039

KEYWORD

nonn,fini,full

AUTHOR

Paolo P. Lava, Mar 06 2013

STATUS

approved

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Last modified February 25 19:39 EST 2021. Contains 341618 sequences. (Running on oeis4.)