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A316630 Numbers k such that 10k+1, 10k+3, 10k+7, and 10k+9 are all composite, and k == 1 (mod 3).

%I A316630 #52 Feb 24 2024 13:52:07
%S A316630 133,196,232,256,298,328,397,403,406,418,430,457,484,640,643,664,709,
%T A316630 727,742,802,847,865,898,907,970,991,1012,1054,1057,1081,1087,1096,
%U A316630 1120,1153,1156,1213,1231,1246,1327,1354,1360,1381,1411,1423,1426,1435,1480,1504
%N A316630 Numbers k such that 10k+1, 10k+3, 10k+7, and 10k+9 are all composite, and k == 1 (mod 3).
%C A316630 The sequence contains all numbers of the form 50151*m + 42175. - _Michael B. Porter_, Jul 16 2018
%C A316630 If m is in the sequence, then so is m + 3*(10*m+1)*(10*m+3)*(10*m+7)*(10*m+9)*k for all k. - _Robert Israel_, Aug 08 2018
%H A316630 Robert Israel, <a href="/A316630/b316630.txt">Table of n, a(n) for n = 1..10000</a>
%H A316630 C. K. Caldwell, The Prime Pages <a href="https://t5k.org/top20/page.php?id=55">Top Twenty: Quadruplet</a>, list of largest known prime quadruplets.
%e A316630 1331 = 11^3, 1333 = 31*43, 1337 = 7*191, 1339 = 13*103, and 133 == 1 (mod 3), so 133 is a sequence member.
%p A316630 remove(t -> ormap(isprime, [10*t+1,10*t+3,10*t+7,10*t+9]), [seq(k,k=1..2000,3)]); # _Robert Israel_, Aug 08 2018
%t A316630 Select[1 + 3 Range@510, Union[ PrimeQ[10 # + {1, 3, 7, 9}]] == {False} &] (* _Robert G. Wilson v_, Jul 16 2018 *)
%t A316630 Select[Range[1,2000,3],AllTrue[10#+{1,3,7,9},CompositeQ]&]  (* _Harvey P. Dale_, Feb 24 2024 *)
%o A316630 (PARI) ok(k)={if(k%3==1, for(i=0, 4, if(isprime(10*k+2*i+1), return(0))); 1, 0)} \\ _Andrew Howroyd_, Jul 10 2018
%o A316630 (MATLAB) m=1; for s=1:510 v=[30*s+11,30*s+13,30*s+17,30*s+19]; if isprime(v)==0  sol(m)=3*s+1; m=m+1;end; end; sol % _Marius A. Burtea_, Sep 17 2019
%o A316630 (Magma) [3*s+1: s in [0..510] | forall{30*s+k: k in [11, 13, 17, 19] | not IsPrime(30*s+k)}]; // _Marius A. Burtea_, Sep 17 2019
%Y A316630 Cf. A007811, A032352.
%K A316630 nonn,easy
%O A316630 1,1
%A A316630 _Patrick A. Thomas_, Jul 09 2018