A023251
Primes that remain prime through 2 iterations of function f(x) = 4x + 9.
Original entry on oeis.org
7, 41, 43, 47, 67, 71, 97, 103, 137, 263, 293, 307, 397, 421, 467, 491, 571, 587, 593, 683, 727, 757, 883, 907, 1021, 1061, 1063, 1097, 1153, 1373, 1427, 1433, 1453, 1523, 1567, 1657, 1747, 1811, 1867, 2141, 2251, 2281, 2287, 2647, 2693, 2791, 2797, 2857, 2927
Offset: 1
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[n: n in [0..100000] | IsPrime(n) and IsPrime(4*n+9) and IsPrime(16*n+45)]; // Vincenzo Librandi, Aug 04 2010
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rpQ[n_]:=AllTrue[Rest[NestList[4#+9&,n,2]],PrimeQ]; Select[Prime[ Range[ 500]],rpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 03 2019 *)
A023282
Primes that remain prime through 3 iterations of function f(x) = 4x + 9.
Original entry on oeis.org
71, 97, 103, 263, 883, 907, 1747, 1867, 2251, 2281, 2693, 2791, 2857, 3067, 3541, 4073, 4513, 4597, 4663, 4793, 6047, 6971, 6983, 8761, 9091, 9203, 9311, 9377, 10343, 11131, 11437, 12037, 12107, 12401, 13451, 13627, 14887, 15881, 16217, 16301, 16493, 16871
Offset: 1
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[n: n in [1..150000] | IsPrime(n) and IsPrime(4*n+9) and IsPrime(16*n+45) and IsPrime(64*n+189)] // Vincenzo Librandi, Aug 04 2010
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Select[Prime@ Range@ 2100, Times @@ Boole@ PrimeQ@ Rest@ NestList[4 # + 9 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 20 2016 *)
A023312
Primes that remain prime through 4 iterations of function f(x) = 4x + 9.
Original entry on oeis.org
883, 2857, 4073, 4663, 9311, 11131, 16493, 18257, 19541, 22063, 28687, 35837, 48383, 55817, 59393, 62131, 71387, 73037, 73133, 78173, 83423, 86111, 88261, 90511, 93287, 93811, 99377, 101051, 104537, 122203, 125927, 149497, 152377, 153941, 155653
Offset: 1
A023340
Primes that remain prime through 5 iterations of function f(x) = 4x + 9.
Original entry on oeis.org
18257, 19541, 22063, 48383, 73037, 73133, 78173, 88261, 93811, 101051, 104537, 152377, 153941, 162343, 168043, 175523, 204251, 211681, 238463, 287341, 375253, 386713, 423961, 659513, 672181, 676103, 688027, 741283, 759491, 770951, 786673
Offset: 1
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[n: n in [1..10000000] | IsPrime(n) and IsPrime(4*n+9) and IsPrime(16*n+45) and IsPrime(64*n+189) and IsPrime(256*n+765) and IsPrime(1024*n+3069)] // Vincenzo Librandi, Aug 05 2010
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is(n)=isprime(n) && isprime(4*n+9) && isprime(16*n+45) && isprime(64*n+189) && isprime(256*n+765) && isprime(1024*n+3069) \\ Charles R Greathouse IV, Oct 11 2016
A153464
Numbers k such that 4*k+9 is not prime.
Original entry on oeis.org
0, 3, 4, 6, 9, 10, 12, 14, 15, 17, 18, 19, 21, 24, 27, 28, 29, 30, 31, 33, 34, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 53, 54, 57, 59, 60, 61, 63, 64, 66, 69, 70, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 87, 88, 89, 90, 92, 93, 94, 96, 99, 101, 102, 104, 105
Offset: 1
Distribution of the terms in the following triangular array:
0;
*,4;
3,*,10;
*,9,*,18;
6,*,17,*,28;
*,14,*,27,*,40;
9,*,24,*,39,*,54;
*,19,*,36,*,53,*,70;
12,*,31,*,50,*,69,*,88;
*,24,*,45,*,66,*,87,*,108;
15,*,38,*,61,*,84,*,107,*,130; etc.
where * marks the non-integer values of (2*h*k + k + h - 4)/2 with h >= k >= 1. - _Vincenzo Librandi_, Jan 14 2013
A378964
a(n) is the least prime of the form p^2 + n*q^2 where p and q are primes, or -1 if there are none.
Original entry on oeis.org
13, 17, 31, 41, 29, 73, 37, 41, 61, 89, 53, 73, 61, 151, 109, 73, 157, 97, 101, 89, 109, 97, 101, 241, 109, 113, 157, 137, -1, 241, 149, 137, 157, 257, 149, 193, 157, 367, 181, 281, 173, 193, 181, 421, 229, 193, 197, 241, 317, 499, 229, 233, -1, 241, 229, 233, 277, 241, -1, 409, 269, 257, 277
Offset: 1
a(6) = 73 because 73 = 7^2 + 6 * 2^2 is the least prime of the form p^2 + 6 * q^2.
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f:= proc(n) uses priqueue;
local pq, p, t;
if n mod 6 = 5 then
if isprime(3^2 + n * 2^2) then return 3^2 + n*2^2
elif isprime(2^2 + n*3^2) then return 2^2 + n*3^2
else return -1
fi fi;
initialize(pq);
insert([-9 - 4*n,3,2],pq);
insert([-4 - 9*n,2,3],pq);
if n mod 3 = 2 then
do
t:= extract(pq);
if isprime(-t[1]) then return -t[1] fi;
if t[2] = 3 then p:= nextprime(t[3]); if p = 3 then p:= 5 fi; insert([-9 - n*p^2,3,p],pq) fi;
if t[3] = 3 then p:= nextprime(t[2]); if p = 3 then p:= 5 fi; insert([-p^2 - n*9,p,3],pq) fi;
od
else
do
t:= extract(pq);
if isprime(-t[1]) then return -t[1] fi;
if t[3] = 2 then p:= nextprime(t[2]); insert([-p^2 - n*4 , p, 2],pq) fi;
if t[2] = 2 or n::even then
p:= nextprime(t[3]); insert([-t[2]^2 - n*p^2,t[2],p],pq)
fi;
od
fi
end proc:
map(f, [$1..200]);
Showing 1-6 of 6 results.
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