A354376 -id:A354376 - cp's OEIS frontend

A354377 Initial terms associated with the arithmetic progressions of primes of A354376.

2, 2, 3, 7, 5, 7, 7, 881, 3499, 199, 75307, 110437, 4943, 31385539, 115453391, 53297929, 3430751869, 4808316343, 8297644387, 214861583621, 5749146449311

Offset: 1

Bernard Schott, May 26 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) = first term i.
The adverb "exactly" requires both i-d and i+n*d to be nonprime (see A113827).
For the corresponding values of the last term, see A354376.
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) != A113827(n) for n = 4, 8, 9, 11. - Michael S. Branicky, May 26 2022

			The first few corresponding arithmetic progressions are:
n = 1 (2);
n = 2 (2, 3);
n = 3 (3, 5, 7);
n = 4 (7, 19, 31, 43);
n = 5 (5, 11, 17, 23, 29);
n = 6 (7, 37, 67, 97, 127, 157);
n = 7 (7, 157, 307, 457, 607, 757, 907)...

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

Cf. A006560, A113827, A354376.

a(8)-a(21) from Michael S. Branicky, May 26 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

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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A354377 Initial terms associated with the arithmetic progressions of primes of A354376.

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

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