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

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

Offset

1,1

Comments

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.

References

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

Links

Table of n, a(n) for n=1..18.

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.

Example

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).

Crossrefs

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

Sequence in context: A238399 A159611 A156585 * A299923 A337189 A087358

Adjacent sequences: A354741 A354742 A354743 * A354745 A354746 A354747

Keyword

nonn,more

Author

Bernard Schott, Jun 05 2022

Status

approved