A354377 -id:A354377 - 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

A354485 Triangle read by rows: row n gives the arithmetic progression of exactly n primes with minimal final term, cf. A354376.

Original entry on oeis.org

2, 2, 3, 3, 5, 7, 7, 19, 31, 43, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 881, 1091, 1301, 1511, 1721, 1931, 2141, 2351, 3499, 3709, 3919, 4129, 4339, 4549, 4759, 4969, 5179, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089

Offset: 1

Bernard Schott, May 29 2022

For the corresponding values of the first term, the last term and the common difference of these arithmetic progressions, see respectively A354377, A354376 and A354484.
Without "exactly", we get A133277.
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 progressions is A006560(n).)

			Triangle begins:
    2;
    2,    3;
    3,    5,    7;
    7,   19,   31,   43;
    5,   11,   17,   23,   29;
    7,   37,   67,   97,  127,  157;
    7,  157,  307,  457,  607,  757,  907;
  881, 1091, 1301, 1511, 1721, 1931, 2141, 2351;
  ...

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

Michael S. Branicky, Table of n, a(n) for n = 1..231 (using A354376 and A354377)

Cf. A006560, A133277, A354376, A354377, A354484.

T(n, 1) = A354377.

T(n, n) = A354376.

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

A354743 Smallest first term of arithmetic progression of exactly n primes with difference A033188(n).

Original entry on oeis.org

2, 2, 3, 41, 5, 7, 7, 881, 3499, 199, 60858179, 147692845283, 14933623, 834172298383, 894476585908771, 1275290173428391, 259268961766921, 1027994118833642281

Offset: 1

Bernard Schott, Jun 05 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the first one which minimizes the common difference d; then a(n) = i.
The word "exactly" requires both i-d and i+n*d to be nonprime.
For the corresponding values of the last term, see A354744.
Without "exactly", we get A033189.
The primes in these arithmetic progressions need not be consecutive.
a(n) != A033189(n) for n = 4, 8, 9, 19 because in these particular cases A113872(n) + A033188(n) is prime.
a(8) = 881 and a(9) = 3499 found by Michael S. Branicky come from A354377.
a(19) > A033189(19) = 1424014323012131633 is not known, it is the smallest first term of an arithmetic progression of exactly 19 primes with a common difference d = 9699690; then a(20) = 1424014323012131633 and a(21) = 28112131522731197609.

			The first few corresponding arithmetic progressions are:
n = 1 and d = 0:   (2);
n = 2 and d = 1:   (2, 3);
n = 3 and d = 2:   (3, 5, 7);
n = 4 and d = 6:   (41, 47, 53, 59);
n = 5 and d = 6:   (5, 11, 17, 23, 29);
n = 6 and d = 30:  (7, 37, 67, 97, 127, 157);
n = 7 and d = 150: (7, 157, 307, 457, 607, 757, 907);
n = 8 and d = 210: (881, 1091,1301, 1511, 1721, 1931, 2141, 2351).

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

J. K. Andersen, Primes in Arithmetic Progression Records.
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A033188, A033189, A113872, A354377, A354744.

A354744 Last term of arithmetic progression of exactly n primes with difference A033188(n) and first term = A354743(n).

Original entry on oeis.org

2, 3, 7, 59, 29, 157, 907, 2351, 5179, 2089, 60881279, 147692870693, 15293983, 834172688773, 894476586329191, 1275290173878841, 259268969935081, 1027994118842320951

Offset: 1

Bernard Schott, Jun 05 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the first one which minimizes the common difference d; then a(n) = i+(n-1)d.
The word "exactly" requires both i-d and i+n*d to be nonprime.
Without "exactly", we get A113872.
The primes in these arithmetic progressions need not be consecutive.
a(n) != 113872(n) for n = 4, 8, 9, 19 because in these particular cases A113872(n) + A033188(n) is prime.
a(8) = 2351 and a(9) = 5179, found by Michael S. Branicky come from A354376.
a(19) > A113872(19) = 1424014323186726053 is not known, it is the last term of the arithmetic progression of exactly 19 primes with a common difference d = 9699690 and first term = A354743(19); then a(20) = 1424014323196425743 and a(21) = 28112131522925191409.

			The first few corresponding arithmetic progressions are:
n = 1 and d = 0:   (2);
n = 2 and d = 1:   (2, 3);
n = 3 and d = 2:   (3, 5, 7);
n = 4 and d = 6:   (41, 47, 53, 59);
n = 5 and d = 6:   (5, 11, 17, 23, 29);
n = 6 and d = 30:  (7, 37, 67, 97, 127, 157);
n = 7 and d = 150: (7, 157, 307, 457, 607, 757, 907);
n = 8 and d = 210: (881, 1091,1301, 1511, 1721, 1931, 2141, 2351).

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

J. K. Andersen, Primes in Arithmetic Progression Records.
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A033188, A033189, A113872, A354377, A354743.

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

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A354485 Triangle read by rows: row n gives the arithmetic progression of exactly n primes with minimal final term, cf. A354376.

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

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A354743 Smallest first term of arithmetic progression of exactly n primes with difference A033188(n).

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A354744 Last term of arithmetic progression of exactly n primes with difference A033188(n) and first term = A354743(n).

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