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Primes which start a run of at least length 2 of consecutive primes == 1 (mod 4).
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%I #15 May 04 2019 00:36:57

%S 13,37,89,109,193,229,277,313,349,389,449,509,613,661,701,757,797,853,

%T 877,929,997,1093,1109,1193,1237,1297,1373,1429,1489,1549,1597,1609,

%U 1637,1669,1709,1733,1789,1873,1889,1933,1993,2069,2113,2137,2153,2213,2269

%N Primes which start a run of at least length 2 of consecutive primes == 1 (mod 4).

%D Enoch Haga, Exploring Primes on Your PC and the Internet, 1994-2007, pp. 30-31. ISBN 978-1-885794-24-6

%H Harvey P. Dale, <a href="/A145990/b145990.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1)=13 because this sequence includes consecutive runs of any length and this first term > 1 in a run of 2 is 13.

%p for i from 2 to 300 do

%p if (ithprime(i) mod 4) = 1 and ithprime(i-1) mod 4 <> 1 and ithprime(i+1) mod 4 = 1 then

%p printf("%d,",ithprime(i)) ;

%p end if;

%p end do: # _R. J. Mathar_, Sep 30 2011

%t Prime[#+1]&/@(SequencePosition[Table[If[Mod[n,4]==1,1,0],{n,Prime[ Range[ 350]]}],{0,1,1},Overlaps->False][[All,1]]) (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 02 2017 *)

%o (UBASIC)

%o 10 'cluster primes

%o 20 C=1

%o 30 input "end #";L

%o 40 for N=3 to L step 2

%o 50 S=int(sqrt(N))

%o 60 for A=3 to S step 2

%o 70 B=N/A

%o 80 if int(B)*A=N then cancel for:goto 170

%o 90 next A

%o 100 C=C+1

%o 110 E=N/4:E=int(E):R=N-(4*E)

%o 120 if R=1 then print N;:C1=C1+1:T1=T1+1:print T1

%o 130 if R=3 then T1=0

%o 140 if R=3 then print " ";N;:C3=C3+1:T2=T2+1:print T2

%o 150 if R=1 then T2=0

%o 160 if T1>10 or T2>10 then stop

%o 170 next

%o 180 print "Total primes=";C;:print "Type A";C1;"Type B";C3

%Y Cf. A039702, A055623, A054624, A145986, A145988 - A145994.

%K easy,nonn

%O 1,1

%A _Enoch Haga_, Oct 26 2008

%E Corrected and extended by _Harvey P. Dale_, Aug 02 2017