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

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

Offset: 1

Author

Bernard Schott, Jun 05 2022

Keywords

• nonn
• more

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.

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.

Examples

			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).
		

References

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

Links

• 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.

Crossrefs

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