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%I #59 Jan 25 2024 07:41:24
%S 121174811,1128318991,2201579179,2715239543,2840465567,3510848161,
%T 3688067693,3893783651,5089850089,5825680093,6649068043,6778294049,
%U 7064865859,7912975891,8099786711,9010802341,9327115723,9491161423,9544001791,10101930253,10523406343,13193702321
%N Initial primes of sets of 6 consecutive primes in arithmetic progression.
%C For all the terms listed so far, the common difference is equal to 30. These are the smallest such sets.
%C It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000 the record is 10 primes.
%C All terms are congruent to 9 (mod 14). - _Zak Seidov_, May 03 2017
%C 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
%C 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
%H Zak Seidov, <a href="/A058362/b058362.txt">Table of n, a(n) for n = 1..102</a>
%H Jens K. Andersen, <a href="http://primerecords.dk/cpap.htm">The Largest Known CPAP's</a>, updated Sep 2018.
%H OEIS Wiki, <a href="/wiki/Consecutive_primes_in_arithmetic_progression">Consecutive primes in arithmetic progression</a>, updated Jan 2020.
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F Equals { A059044(i) | A059044(i+1) = A151800(A059044(i)) }, A151800 = nextprime. - _M. F. Hasler_, Oct 30 2018
%o (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
%Y Cf. A006560: first prime to start a CPAP-n.
%Y Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
%Y Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
%Y Cf. A052239: starting prime of first CPAP-4 with common difference 6n.
%Y Cf. A059044: starting primes of CPAP-5.
%Y Cf. A210727: starting primes of CPAP-5 with common difference 60.
%K nonn
%O 1,1
%A Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000
%E Corrected by _Jud McCranie_, Jan 04 2001
%E a(11)-a(18) from _Donovan Johnson_, Sep 05 2008
%E Comment split off from Name (to clarify definition) by _M. F. Hasler_, Oct 27 2018
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