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

133, 196, 232, 256, 298, 328, 397, 403, 406, 418, 430, 457, 484, 640, 643, 664, 709, 727, 742, 802, 847, 865, 898, 907, 970, 991, 1012, 1054, 1057, 1081, 1087, 1096, 1120, 1153, 1156, 1213, 1231, 1246, 1327, 1354, 1360, 1381, 1411, 1423, 1426, 1435, 1480, 1504

Offset: 1

Author

Patrick A. Thomas, Jul 09 2018

Keywords

• nonn
• easy

Comments

The sequence contains all numbers of the form 50151*m + 42175. - Michael B. Porter, Jul 16 2018

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

Examples

			1331 = 11^3, 1333 = 31*43, 1337 = 7*191, 1339 = 13*103, and 133 == 1 (mod 3), so 133 is a sequence member.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• C. K. Caldwell, The Prime Pages Top Twenty: Quadruplet, list of largest known prime quadruplets.

Crossrefs

Cf. A007811, A032352.

Programs

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

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

Maple

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

Mathematica

Select[1 + 3 Range@510, Union[ PrimeQ[10 # + {1, 3, 7, 9}]] == {False} &] (* Robert G. Wilson v, Jul 16 2018 *)
Select[Range[1,2000,3],AllTrue[10#+{1,3,7,9},CompositeQ]&]  (* Harvey P. Dale, Feb 24 2024 *)

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