cp's OEIS Frontend

All terms are prime, since this is necessary and sufficient to get a prime for x = 0.

The values given in A273770 are the number of consecutive primes obtained for x = 0, 1, 2, ....

Sequence A273595 is the subsequence of terms for which 2n+1 is prime.

For even coefficients of the linear term, the answer would always be q=2, the only choice that yields a prime for x=0 and also for x=1 if (coefficient of the linear term)+3 is prime.

The initial term a(n=0) = 41 corresponds to Euler's famous prime-generating polynomial 41+x+x^2. Some subsequent terms are equal to the primes this polynomial takes for x=1,2,3,.... This stems from the fact that adding 2 to the coefficient of the linear term is equivalent to shifting the x-variable by 1. Since here we require x >= 0, we find a reduced subset of the previous sequence of primes, missing the first one, starting with q equal to the second one. (It is known that there is no better prime-generating polynomial of this form than Euler's, see the MathWorld page and A014556. "Better" means a larger p producing p-1 primes in a row. However, the prime k-tuple conjecture suggests that there should be arbitrarily long runs of primes of this form (for much larger p), i.e., longer than 41, but certainly much less than the respective p. Therefore we speak of local maxima.)

It is well known that for 0 <= n <= 40, the polynomial f(n) = n^2 - n + 41 does yield a prime number, so the sequence is well defined.

A variant of A005846, A060566, A142719. All these aim at extending the series of prime values of Euler's famous prime-producing polynomial P(n) = n^2 + n + 41, see references in A005846. [The present sequence considers f(n) = P(n-1) which is completely equivalent.]

The present sequence is a simplification of an extended variant of A142719. By construction, all terms of the present sequence are prime, but in contrast to A005846, prime values of the polynomial remain at the "correct" position, a(n) = f(n). The "substituted" values are easily recognized as they follow local maxima. Of course one could equally well insert a(n) = 2 whenever f(n) is composite.

The present sequence contains only primes. A different sequence, defined by "a(n) = f(n) if this is prime, a(n) = f(n-40) otherwise", does not always produce primes.

Integers, in order (up to and including p. 36) appearing in the 1973 novel "Gravity's Rainbow".

Rules: In order to avoid confusion with rational numbers, numbers in strings less representative of entire isolated integers (e.g. "6:43:16 BDST", p. 29), single-quotation-mark-truncated integers (e.g. " '37 ", p. 20, as opposed to "1937"), etc., we begin by establishing the following rules:

(i) The integers cannot be connected to non-space-characters except if: the integer is immediately followed by a period then a space, or a comma then a space, or a question mark then a space, or an exclamation point then a space, or a colon then a space, or a semicolon then a space, or a closing bracket, or a double quotation mark (but not a single quotation mark, e.g. "86'd" p. 22); or except if an opening bracket immediately precedes the integer, or if a double quotation mark (but not a single quotation mark, given the truncation concern indicated above) immediately precedes the integer; integers may be concatenated using "-".

(ii) Integers containing a comma such as "50,000" (p. 85) are allowed.

(iii) These numbers cannot be written using characters of the alphabet, e.g. "Zero" (p. 1), or "World War II" (p. 17).

(iv) The page numbers and chapter numbers of the book are not included (although the corresponding page numbers are indicated below).

(v) Integers appearing in equations are included.

(End of Rules)

See examples below for integers which are and are not(/would not be) included by the above rules.

The corresponding page numbers (in the below edition) are: 3, 7, 7, 9, 14, 17, 18, 21, 21, 25, 27, 27, 27, 28, 29, 29, 31, 34, 34, 36, 36, 36.