A354485 -id:A354485 - cp's OEIS frontend

A354376 Smallest prime which is at the end of an arithmetic progression of exactly n primes.

2, 3, 7, 43, 29, 157, 907, 2351, 5179, 2089, 375607, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111

Offset: 1

Bernard Schott, May 24 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.
The word "exactly" requires both i-d and i+n*d to be nonprime; without "exactly", we get A005115.
For the corresponding values of the first term, and the common difference, see A354377 and A354484. For the actual arithmetic progressions, see A354485.
The primes in these arithmetic progressions need not be consecutive. (The smallest prime at the start of a run of exactly n consecutive primes in arithmetic progression is A006560(n).)
a(n) != A005115(n), because A005115(n) + A093364(n) is prime for n = 4, 8, 9, 11. - Michael S. Branicky, May 24 2022

			The arithmetic progression (5, 11, 17, 23) with common difference 6 contains 4 primes, but 29 = 23+6 is also prime, hence a(4) != 23.
The arithmetic progression (7, 19, 31, 43) with common difference 12 also contains 4 primes, and 7-12 < 0 and 43+12 = 55 is composite; moreover this arithmetic progression is the smallest such progression with exactly 4 primes, hence a(4) = 43.

R. K. Guy, Unsolved Problems in Number Theory, A5, Arithmetic progressions of primes.

Cf. A005115, A006560, A093364, A354377, A354484, A354485.

from sympy import isprime, nextprime
def a(n):
    if n < 3: return [2, 3][n-1]
    p = 2
    while True:
        for d in range(2, (p-3)//(n-1)+1, 2):
            if isprime(p+d) or isprime(p-n*d): continue
            if all(isprime(p-j*d) for j in range(1, n)): return p
        p = nextprime(p)
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, May 24 2022

a(4) corrected and a(8)-a(13) from Michael S. Branicky, May 24 2022

a(14)-a(21) derived using A005115 and A093364 by Michael S. Branicky, May 24 2022

A354484 Common differences associated with the arithmetic progressions of primes in A354376.

Original entry on oeis.org

0, 1, 2, 12, 6, 30, 150, 210, 210, 210, 30030, 13860, 60060, 420420, 4144140, 9699690, 87297210, 717777060, 4180566390, 18846497670, 26004868890

Offset: 1

Bernard Schott, May 28 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the one which minimizes the last term; then a(n) = common difference d.
The word "exactly" requires both i-d and i+n*d to be nonprime; without "exactly", we get A093364.
For the corresponding values of the first term and the last term, see respectively A354377 and A354376. For the actual arithmetic progressions, see A354485.
The primes in these arithmetic progressions need not be consecutive. (The smallest prime at the start of a run of exactly n consecutive primes in arithmetic progression is A006560(n).)

			The first few corresponding arithmetic progressions are:
d = 0:   (2);
d = 1:   (2, 3);
d = 2:   (3, 5, 7);
d = 12:  (7, 19, 31, 43);
d = 6:   (5, 11, 17, 23, 29);
d = 30:  (7, 37, 67, 97, 127, 157);
d = 150: (7, 157, 307, 457, 607, 757, 907).

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A5, Arithmetic progressions of primes, pp. 25-28.

Cf. A006560, A093364, A354376, A354377, A354485.

a(1) = 0, then for n > 1, a(n) = (A354376(n) - A354377(n)) / (n-1).

a(7)-a(21) via A354376, A354377 from Michael S. Branicky, May 28 2022

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A354376 Smallest prime which is at the end of an arithmetic progression of exactly n primes.

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