A093364 -id:A093364 - cp's OEIS frontend

A126989 Gaps associated with the first and smallest arithmetic progressions of n consecutive primes in A006560.

Original entry on oeis.org

0, 1, 2, 6, 30, 30, 210

Offset: 1

Artur Jasinski, Jan 01 2007

The gap for the first and smallest AP of 7 consecutive primes is at least 210 (so the 7th term is not definitive).

P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004.

H. Dubner et al., Ten consecutive primes in arithmetic progression, Mathematics of Computation, Volume 71, Number 239, Pages 1323-1328.

Cf. A006560, A005115, A093364.

a(7) corrected by Stephen Tucker, Jan 25 2009

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

A093366 Gaps associated with the arithmetic progressions in A093365.

Original entry on oeis.org

0, 3, 3, 8, 8, 12, 24, 24, 24, 24, 24, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 2772, 21252, 21252, 21252, 42504, 42504, 63756, 63756, 63756, 825132, 825132, 825132

Offset: 1

Hugo Pfoertner, Apr 27 2004

Cf. A000404, A005115, A093364, A093365.

Erroneous terms a(31) and a(32) corrected by Hugo Pfoertner, Oct 15 2010

A122764 Initial terms of arithmetic progression of primes in A005115 with duplicates removed.

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, Sep 22 2006

Old title was: "In A005115 a+b*j: the a term with duplicates removed."

If duplicate terms are not removed, we obtain A113827. - Charlie Neder, Feb 02 2019

			A005115(7) comes from the 7-term prime progression {7, 157, 307, 457, 607, 757, 907}, and so 7 is in this sequence. - _Charlie Neder_, Feb 02 2019

Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Pages 481-547 from Volume 167 (2008), Issue 2.

Cf. A093364, A005115.

a = {{ 1, 2, 2}, {2, 2 + j, 3}, {3, 3 + 2j, 7}, {4, 5 + 6j, 23}, {5, 5 + 6j, 29}, {6, 7 + 30j, 157}, {7, 7 + 150j, 907}, {8, 199 + 210j, 1669}, {9, 199 + 210j, 1879}, {10, 199 + 210j, 2089}, {11, 110437 + 13860j, 249037}, {12, 110437 + 13860j, 262897}}
Union[Table[CoefficientList[a[[n, 2]], j][[1]], {n, 1, 12}]]

Better name and a(7)-a(15) from Charlie Neder, Feb 02 2019

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A126989 Gaps associated with the first and smallest arithmetic progressions of n consecutive primes in A006560.

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

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A093366 Gaps associated with the arithmetic progressions in A093365.

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A122764 Initial terms of arithmetic progression of primes in A005115 with duplicates removed.

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