A354499 Number of consecutive primes generated by adding 2n to the odd squares (A016754).

2, 4, 1, 0, 2, 1, 0, 1, 1, 0, 5, 0, 0, 3, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 14, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 4, 0, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 8, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 1, 0

Offset

1,1

Comments

Conjecture: a(n) <= 18 = a(326).

a(m) = 0 for m in A047845. - Michel Marcus, Aug 16 2022

I conjecture the opposite: a(n) is unbounded, and indeed for any k < 1 and any m there are >> x^k terms up to x with a(n) > m. At a very rough guess, there should be some n with 20-50 digits having a(n) > 18. - Charles R Greathouse IV, Oct 26 2022

Links

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

Formula

a(n) is number of consecutive primes generated by (2x-1)^2+2n for x=1,2,3,4,

Example

For n=1 we have 1^2+2*1=3 and 3^2+2*1=11 are prime but 5^2+2*1=27 is not, and thus a(1)=2.
For n=2, 1^2+2*2=5 ... 7^2+2*2=53 are prime but 9^2+2*2=85 is not, thus a(2)=4.
For n=3, 1^2+2*3=7 is prime but 3^2+2*3=15 is not thus a(3)=1.
For n=4, 1^2+2*4=9 which is not prime, thus a(4)=0.

Maple

f:= proc(n) local k;
  for k from 1 by 2 do
    if not isprime(k^2+2*n) then return (k-1)/2 fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Oct 26 2023

Mathematica

a[n_] := Module[{k = 1}, While[PrimeQ[k^2 + 2*n], k += 2]; (k - 1)/2]; Array[a, 100] (* Amiram Eldar, Aug 15 2022 *)

Prog

(PARI) a(n) = my(k=1); while (isprime(k^2+2*n), k+=2); (k-1)/2; \\ Michel Marcus, Aug 16 2022

Crossrefs

Cf. A005843 (even numbers), A016754 (odd squares), A356567 (positions of records).

Cf. A047845.

Sequence in context: A055599 A115407 A010586 * A070678 A124091 A067849

Adjacent sequences: A354496 A354497 A354498 * A354500 A354501 A354502

Keyword

nonn

Author

Steven M. Altschuld, Aug 15 2022

Status

approved