A058362 Initial primes of sets of 6 consecutive primes in arithmetic progression.

121174811, 1128318991, 2201579179, 2715239543, 2840465567, 3510848161, 3688067693, 3893783651, 5089850089, 5825680093, 6649068043, 6778294049, 7064865859, 7912975891, 8099786711, 9010802341, 9327115723, 9491161423, 9544001791, 10101930253, 10523406343, 13193702321

Offset: 1

Author

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

Keywords

• nonn

Comments

For all the terms listed so far, the common difference is equal to 30. These are the smallest such sets.

It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000 the record is 10 primes.

All terms are congruent to 9 (mod 14). - Zak Seidov, May 03 2017

The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3*10^11, cf. A210727. [With a slope of a(n)/n ~ 5*10^8 this would correspond to n ~ 600.] This sequence consists of first members of pairs of consecutive primes in A059044. Conversely, a pair of consecutive primes in this sequence starts a CPAP-7. This must have a common difference >= 210. As of today, the smallest known CPAP-7 starts at 382003672700092872707633 ~ 3.8*10^23, cf. Andersen link. - M. F. Hasler, Oct 27 2018

The common difference of 60 first occurs at a larger-than-expected prime. The first CPAP-6 with common difference 90 starts at 8560443932347. The first CPAP-6 with common difference 120 starts at 1925601119017087. - Jerry M Lagrou, Jan 01 2024

Links

• Zak Seidov, Table of n, a(n) for n = 1..102
• Jens K. Andersen, The Largest Known CPAP's, updated Sep 2018.
• OEIS Wiki, Consecutive primes in arithmetic progression, updated Jan 2020.
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Cf. A006560: first prime to start a CPAP-n.

Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.

Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).

Cf. A052239: starting prime of first CPAP-4 with common difference 6n.

Cf. A059044: starting primes of CPAP-5.

Cf. A210727: starting primes of CPAP-5 with common difference 60.

Programs

PARI

p=c=g=P=0;forprime(q=1,, p+g==(p+=g=q-p)|| next; q==P+2*g&& c++|| c=3; c>5&& print1(P-3*g,","); P=q-g) \\ M. F. Hasler, Oct 26 2018

Formula

Equals { A059044(i) | A059044(i+1) = A151800(A059044(i)) }, A151800 = nextprime. - M. F. Hasler, Oct 30 2018

Extensions

Corrected by Jud McCranie, Jan 04 2001

a(11)-a(18) from Donovan Johnson, Sep 05 2008

Comment split off from Name (to clarify definition) by M. F. Hasler, Oct 27 2018