A052239 - OEIS

%I #41 Dec 03 2023 01:24:42

%S 251,111497,74453,1397609,642427,5321191,23921257,55410683,400948369,

%T 253444777,1140813701,491525857,998051413,2060959049,4480114337,

%U 55140921491,38415872947,315392068463,15162919459,60600021611,278300877401,477836574947,1486135570643

%N Smallest prime p in set of 4 consecutive primes in arithmetic progression with common difference 6n.

%C See also the less restrictive A054701 where the gaps are multiples 6n. - _M. F. Hasler_, Nov 06 2018

%H Jerry M Lagrou, <a href="/A052239/b052239.txt">Table of n, a(n) for n = 1..42</a>

%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>

%e a(5) = 642427, 642457, 642487, 642517 are the smallest consecutive primes with 3 consecutive gaps of 30, cf. A052243.

%e From _M. F. Hasler_, Nov 06 2018: (Start)

%e Other terms are also initial terms of corresponding sequences:

%e a(1) = 251 = A033451(1) = A054800(1), start of first CPAP-4 with common gap of 6,

%e a(2) = 111497 = A033447(1), start of first CPAP-4 with common gap of 12,

%e a(3) = 74453 = A033448(1) = A054800(25), first CPAP-4 with common gap of 18,

%e a(4) = 1397609 = A052242(1), start of first CPAP-4 with common gap of 24,

%e a(5) = 642427 = A052243(1) = A052195(16), first CPAP-4 with common gap of 30,

%e a(6) = 5321191 = A058252(1) = A161534(26), first CPAP-4 with common gap 36 = 6^2,

%e a(7) = 23921257 = A058323(1), start of first CPAP-4 with common gap of 42,

%e a(8) = 55410683 = A067388(1), start of first CPAP-4 with common gap of 48,

%e a(9) = 400948369 = A259224(1), start of first CPAP-4 with common gap of 54,

%e a(10) = 253444777 = A210683(1) = A089234(417), CPAP-4 with common gap of 60,

%e a(11) = 1140813701 = A287547(1), start of first CPAP-4 with common gap of 66,

%e a(12) = 491525857 = A287550(1), start of first CPAP-4 with common gap of 72,

%e a(13) = 998051413 = A287171(1), start of first CPAP-4 with common gap of 78,

%e a(14) = 2060959049 = A287593(1), start of first CPAP-4 with common gap of 84,

%e a(15) = 4480114337 = A286817(1) = A204852(444), common distance 90. (End)

%t Transpose[Flatten[Table[Select[Partition[Prime[Range[2000000]],4,1], Union[ Differences[ #]] =={6n}&,1],{n,7}],1]][[1]] (* _Harvey P. Dale_, Aug 12 2012 *)

%o (PARI) a(n, p=[2, 0, 0], d=6*[n, n, n])={while(p+d!=p=[nextprime(p[1]+1), p[1], p[2]], ); p[3]-d[3]} \\ after _M. F. Hasler_ in A052243; Graziano Aglietti (mg5055(AT)mclink.it), Aug 22 2010, Corrected by _M. F. Hasler_, Nov 06 2018

%o (PARI) A052239(n, p=2, c, o)={n*=6; forprime(q=p+1, , if(p+n!=p=q, next, q!=o+2*n, c=2, c++>3, break); o=q-n); o-n} \\ _M. F. Hasler_, Nov 06 2018

%Y Initial terms of sequences A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388, A259224, A210683.

%Y Range is a subset of A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).

%Y Cf. A054701: gaps are possibly distinct multiples of 6n (not CPAP's).

%K nice,nonn

%O 1,1

%A _Labos Elemer_, Jan 31 2000

%E More terms from _Labos Elemer_, Jan 04 2002

%E a(7) corrected and more terms added by Graziano Aglietti (mg5055(AT)mclink.it), Aug 22 2010

%E a(15)-a(20) from _Donovan Johnson_, Oct 05 2010

%E a(21)-a(23) from _Donovan Johnson_, May 23 2011