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

0, 1, 2, 12, 6, 30, 150, 210, 210, 210, 30030, 13860, 60060, 420420, 4144140, 9699690, 87297210, 717777060, 4180566390, 18846497670, 26004868890

Offset: 1

Author

Bernard Schott, May 28 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 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).)

Examples

			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).
		

References

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

Crossrefs

Cf. A006560, A093364, A354376, A354377, A354485.

Formula

a(1) = 0, then for n > 1, a(n) = (A354376(n) - A354377(n)) / (n-1).

Extensions

a(7)-a(21) via A354376, A354377 from Michael S. Branicky, May 28 2022