A354744 -id:A354744 - cp's OEIS frontend

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

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.

cp's OEIS Frontend

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

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