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A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

%I A059044 #46 Jan 02 2020 13:14:10
%S A059044 9843019,37772429,53868649,71427757,78364549,79080577,98150021,
%T A059044 99591433,104436889,106457509,111267419,121174811,121174841,168236119,
%U A059044 199450099,203908891,207068803,216618187,230952859,234058871,235524781,253412317,263651161,268843033,294485363,296239787
%N A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.
%C A059044 Each set has a constant difference of 30, for all of the terms listed so far.
%C A059044 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 A059044 The first CPAP-5 with common difference 60 starts at 6182296037 ~ 6e9, cf. A210727. This sequence consists of first members of pairs of consecutive primes in A054800 (see also formula): a(1..6) = A054800({1555, 4555, 6123, 7695, 8306, 8371}). Conversely, pairs of consecutive primes in this sequence yield a term of A058362, i.e., they start a sequence of 6 consecutive primes in arithmetic progression (CPAP-6): e.g., the nearby values a(12) = 121174811, a(13) = 121174841 = a(12) + 30 indicate such a term, whence A006560(6) = A058362(1) = a(12). The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3e11, cf. A210727. Longer CPAP's must have common difference >= 210. - _M. F. Hasler_, Oct 26 2018
%C A059044 About 500 initial terms of this sequence are the same as for the sequence "First of 5 consecutive primes separated by gaps of 30". The first 10^4 terms of A052243 give 281 terms of this sequence (up to ~ 3.34e9) with the same formula as the one using A054800, but as the above comment says, this will miss terms beyond twice that range. - _M. F. Hasler_, Jan 02 2020
%D A059044 David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 181.
%H A059044 Zak Seidov, <a href="/A059044/b059044.txt">Table of n, a(n) for n = 1..241</a> (all terms up to 3*10^9)
%H A059044 Jens K. Andersen, <a href="http://primerecords.dk/cpap.htm">The Largest Known CPAP's</a>, updated Sept. 2018
%H A059044 OEIS wiki, <a href="/wiki/Consecutive_primes_in_arithmetic_progression">Consecutive primes in arithmetic progression</a>, updated Jan. 2020
%H A059044 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F A059044 Found by exhaustive search for 5 primes in arithmetic progression with all other intermediate numbers being composite.
%F A059044 A059044 = { A054800(i) | A054800(i+1) - A151800(A054800(i)) } with the nextprime function A151800(prime(k)) = prime(k+1) = prime(k) + A001223(k). - _M. F. Hasler_, Oct 27 2018, edited Jan 02 2020.
%t A059044 Select[Partition[Prime[Range[14000000]],5,1],Length[Union[ Differences[ #]]]==1&] (* _Harvey P. Dale_, Jun 22 2013 *)
%o A059044 (PARI) A059044(n,p=2,c,g,P)={forprime(q=p+1,, if(p+g!=p+=g=q-p, next, q!=P+2*g, c=3, c++>4, print1(P-2*g,",");n--||break);P=q-g);P-2*g} \\ This does not impose the gap to be 30, but it happens to be the case for the first values. - _M. F. Hasler_, Oct 26 2018
%Y A059044 Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
%Y A059044 Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
%Y A059044 Cf. A052239: start of first CPAP-4 with common difference 6n.
%Y A059044 Cf. A058362: start of 6 consecutive primes in arithmetic progression.
%Y A059044 Cf. A006560: first prime to start a CPAP-n.
%K A059044 nonn
%O A059044 1,1
%A A059044 Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000
%E A059044 a(16)-a(22) from _Donovan Johnson_, Sep 05 2008
%E A059044 Reference added by _Harvey P. Dale_, Jun 22 2013
%E A059044 Edited (definition clarified, cross-references corrected and extended) by _M. F. Hasler_, Oct 26 2018