This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(4) = 27 for 27*30 = 810 yields twin primes at 810+11 = A001359(32) = A000040(142) and 810+17 = A001359(33) = A000040(144) ending at 810+19 = A000040(145).
a014561Q[n_Integer] :=
If[And[PrimeQ[30 n + 11], PrimeQ[30 n + 13], PrimeQ[30 n + 17],
PrimeQ[30 n + 19]] == True, True, False];
a014561[n_Integer] :=
Flatten[Position[Thread[a014561Q[Range[n]]], True]];
a014561[1000] (* Michael De Vlieger, Jul 17 2014 *)
Select[Range[0,6000],AllTrue[30#+{11,13,17,19},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 21 2016 *)
isok(n) = isprime(30*n+11) && isprime(30*n+13) && isprime(30*n+17) && isprime(30*n+19) \\ Michel Marcus, Jun 09 2013
[ n: n in [0..6000000] | forall{ q: q in [1, 7, 11, 13, 17, 19, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011
Select[Range[6*10^6],AllTrue[30#+{1,7,11,13,17,19,29},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 21 2021 *)
[ n: n in [0..3000] | forall{ q: q in [1, 7, 11, 13, 17, 19, 23, 29] | not IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011
Select[Range[3000],AllTrue[30#+{1,7,11,13,17,19,23,29},CompositeQ]&] (* Harvey P. Dale, Jan 06 2022 *)
{cav(mx)= local(wp=[1,7,11,13,17,19,23,29],v=[],i,j,m); for(k=1,mx, i=k*30;j=1;m=1;while(m&&(j<9),m=!isprime(i+wp[j]);j+=1);if(m,v=concat(v,k))); return(v)}
a(1)=7 because there are 7 primes in the interval (30,60): 31,37,41,43,47,53,59. a(26)=3 because the interval of length 30 following 26*30=780 contains 3 primes: 787, 797 and 809.
! See links given in A098591.
a(n) = primepi(30*(n+1)) - primepi(30*n); \\ Michel Marcus, Apr 04 2020
from sympy import primerange def a(n): return len(list(primerange(n*30, (n+1)*30))) print([a(n) for n in range(106)]) # Michael S. Branicky, Oct 07 2021
[ n: n in [5..70000000 by 7] | forall{ q: q in [1, 7, 13, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011
filter:= proc(n) local j; andmap(isprime, [seq(30*n+j,j=[1,7,13,17,19,23,29])]) end proc: select(filter, [seq(i,i=5..5*10^6,7)]); # Robert Israel, Nov 04 2024
Select[Range[42*10^5],AllTrue[30#+{1,7,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 10 2018 *)
[ n: n in [4..70000000 by 7] | forall{ q: q in [1, 7, 11, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011
Select[Range[5000000],And@@PrimeQ[30 #+{1,7,11,17,19,23,29}]&] (* Harvey P. Dale, Mar 06 2011 *)
[ n: n in [0..5000000] | forall{ q: q in [1, 7, 11, 13, 17, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 23 2011
a:= proc(n) option remember;
local m;
if n=1 then 1
else for m from 30*(a(n-1)+7) by 210
while not (isprime (m+1) and isprime (m+7) and
isprime (m+11) and isprime (m+13) and
isprime (m+17) and isprime (m+23) and
isprime (m+29))
do od; m/30
fi
end:
seq (a(n), n=1..10);
Select[Range[5000000],And@@PrimeQ/@(30(#)+{1,7,11,13,17,23,29})&] (* Harvey P. Dale, Feb 23 2011 *)
[ n: n in [2..70000000 by 7] | forall{ q: q in [1, 7, 11, 13, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011
Select[Range[7*10^6],AllTrue[30#+{1,7,11,13,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 16 2016 *)
1 is a term since there are 7 primes in 30..60: 31, 37, 41, 43, 47, 53, 59. 2 is a term since there are 7 primes in 60..90: 61, 67, 71, 73, 79, 83, 89. 3 is not a term since there are only 6 primes in 90..120: 97, 101, 103, 107, 109, 113. 49 is a term since there are 7 primes in 30*49..30*50: 1471, 1481, 1483, 1487, 1489, 1493, 1499.
ArrayPlot[Table[Boole@PrimeQ[i*30+j],{i,0,399},{j,30}],Mesh->True]
index=1;Do[If[Length@(*PrimeRange=*) Select[Range[30*k+1,30*k+30,2],PrimeQ]==7,Print[index++," ",k]],{k,1,10^9}]
[n|n<-[1..10^6],#primes([30*n,30*n+30])==7]
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