A284708 Smallest initial prime p for at least n primes in increasing arithmetic progression with a common difference less than p.

2, 2, 3, 11, 37, 107, 409, 409, 409, 25471, 53173, 65003, 766439, 11797483

Offset: 1

Author

Arkadiusz Wesolowski, Jan 09 2018

Keywords

• nonn
• more

Comments

Conjecture: a(n) > A034386(n) for every n >= 4.

From Bernard Schott, Mar 15 2023: (Start)

Corresponding common differences are in A361492.

a(22) = 11410337850553 since it is the smallest term in a sequence of 22 primes in arithmetic progression, and the corresponding common difference 4609098694200 is < a(22) (see Penguin reference). (End)

Examples

			Smallest initial prime p, primes in arithmetic progression:
a(1) = 2: (2);
a(2) = 2: (2, 3);
a(3) = 3: (3, 5, 7);
a(4) = 11: (11, 17, 23, 29);
a(5) = 37: (37, 67, 97, 127, 157);
a(6) = 107: (107, 137, 167, 197, 227, 257);
a(7) = 409: (409, 619, 829, 1039, 1249, 1459, 1669);
a(8) = 409: (409, 619, 829, 1039, 1249, 1459, 1669, 1879);
a(9) = 409: (409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089);
		

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 11410337850553, page 191.

Crossrefs

Cf. A361492.

Cf. A123556, A342309.

Programs

PARI

isokd(p, n, d) = for (i=1, n, if (!isprime(p+(i-1)*d), return(0))); 1;
isokp(p, n) = for (d=1, p-1, if (isokd(p, n, d), return(1)););
a(n) = my(p=2); while (!isokp(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Mar 15 2023

Extensions

Name edited by Bernard Schott, Mar 15 2023