A247867 a(n) is the smallest prime in the interval [k*sqrt(k), k*sqrt(k+2)], where k = A001359(n), or a(n)=0 if there is no prime in this interval.

0, 13, 37, 71, 157, 263, 457, 599, 1019, 1109, 1607, 1823, 2399, 2647, 2767, 3433, 3697, 4421, 4721, 5501, 6469, 8581, 8951, 9901, 11897, 13577, 14669, 15329, 16229, 16921, 23011, 23531, 23789, 25097, 26153, 32531, 33107, 33997, 34583, 36037, 39079, 43093

Offset: 1

Author

Vladimir Shevelev, Sep 25 2014

Keywords

• nonn

Comments

The sequence is partly connected with conjecture in A247834. In turn, we conjecture that all terms a(n)>0 for n>1.

Examples

			For n=1, k=A001359(1)=3, we have the interval [3*sqrt(3), 3*sqrt(5)] = [5.1...,6.7...] which does not contain a prime. So, a(1)=0.
For n=2, k=5, we have the interval [5*sqrt(5), 5*sqrt(7)] = [11.1..., 13.2...] which contains only one prime: 13. So, a(2)=13.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001359, A247834, A247835.

Programs

Maple

p:= 1: q:= 2: count:= 0:
while count < 100 do
  if q = p+2 then
    count:= count+1;
    r:= nextprime(floor(p*sqrt(p)));
    if r^2 < p^2*q then A[count]:= r
    else A[count]:= 0 fi;
  fi;
  p:= q; q:= nextprime(p);
od:
seq(A[i],i=1..100); # Robert Israel, Apr 08 2018

PARI

lista(nn) = {forprime(p=2, nn, if (isprime(q=p+2), pmin = nextprime(ceil(p*sqrt(p))); if (pmin <= floor(p*sqrt(q)), val = pmin, val = 0); print1(val, ", ");););} \\ Michel Marcus, Sep 25 2014

Extensions

More terms from Michel Marcus, Sep 25 2014