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A356567
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Numbers that generate increasing numbers of consecutive primes when doubled and added to the sequence of odd squares. (Positions of records in A354499.)
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1
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OFFSET
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1,2
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COMMENTS
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With a(1)=1, a(n) such that (2*x-1)^2 + 2*a(n) gives prime numbers for x=1 to k where the k for a(n) exceeds the k for a(n-1), a(n-2), ..., a(1).
Conjecture: this list is complete, since primes get farther apart as numbers increase. (2x-1)^2 + 2*29 generates many primes, with 38 of the first 43 and 105 of the first 156 values of x generating primes.
For the (2x-1)^2 + 2*29 values that are not prime, there seems to be a restriction on the factors. No values with prime factors below 29 were seen, nor were 41, 43, 53, 71, 73, 89, 97, ... For each of the other a(n) (or indeed any other natural number K), it seems there is a list of acceptable prime factors for the (2x-1)^2 + 2*K value. This gives a curious connection between addition and prime factors.
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LINKS
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EXAMPLE
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a(1)=1, because 1^2+2*1=3 and 3^2+2*1=11 are prime but 5^2+2*1=27 is not, and thus k=2.
a(2)=2, because 1^2+2*2=5 ... 7^2+2*2=53 are prime but 9^2+2*2=85 is not, thus k=4.
a(3)=11, because 1^2+2*11=23 ... 9^2+2*11=103 are prime, thus k=5.
a(4)=29, because 1^2+2*29=59 ... 27^2+2*29=787 are all prime, thus k=14.
a(5)=326 because 1^2+2*326=653 ... 35^2+2*352=1877 are all prime, thus k=18.
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MATHEMATICA
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f[n_] := Module[{k = 1}, While[PrimeQ[k^2 + 2*n], k += 2]; (k - 1)/2]; s = {}; fm = 0; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^3}]; s (* Amiram Eldar, Aug 15 2022 *)
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PROG
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(PARI) f(n) = my(k=1); while (isprime(k^2+2*n), k+=2); (k-1)/2; \\ A354499
lista(nn) = my(m=0); for (n=1, nn, my(x = f(n)); if (x > m, m = x; print1(n, ", ")); ); \\ Michel Marcus, Aug 16 2022
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CROSSREFS
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Cf. A016754 (odd squares), A354499 (consecutive primes generated by adding 2n to odd squares).
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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