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

%I A354499 #28 Oct 26 2023 20:18:13
%S A354499 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,
%T A354499 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,
%U A354499 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
%N A354499 Number of consecutive primes generated by adding 2n to the odd squares (A016754).
%C A354499 Conjecture: a(n) <= 18 = a(326).
%C A354499 a(m) = 0 for m in A047845. - _Michel Marcus_, Aug 16 2022
%C A354499 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
%H A354499 Robert Israel, <a href="/A354499/b354499.txt">Table of n, a(n) for n = 1..10000</a>
%F A354499 a(n) is number of consecutive primes generated by (2x-1)^2+2n for x=1,2,3,4,
%e A354499 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.
%e A354499 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.
%e A354499 For n=3, 1^2+2*3=7 is prime but 3^2+2*3=15 is not thus a(3)=1.
%e A354499 For n=4, 1^2+2*4=9 which is not prime, thus a(4)=0.
%p A354499 f:= proc(n) local k;
%p A354499   for k from 1 by 2 do
%p A354499     if not isprime(k^2+2*n) then return (k-1)/2 fi
%p A354499   od
%p A354499 end proc:
%p A354499 map(f, [$1..100]); # _Robert Israel_, Oct 26 2023
%t A354499 a[n_] := Module[{k = 1}, While[PrimeQ[k^2 + 2*n], k += 2]; (k - 1)/2]; Array[a, 100] (* _Amiram Eldar_, Aug 15 2022 *)
%o A354499 (PARI) a(n) = my(k=1); while (isprime(k^2+2*n), k+=2); (k-1)/2; \\ _Michel Marcus_, Aug 16 2022
%Y A354499 Cf. A005843 (even numbers), A016754 (odd squares), A356567 (positions of records).
%Y A354499 Cf. A047845.
%K A354499 nonn
%O A354499 1,1
%A A354499 _Steven M. Altschuld_, Aug 15 2022