A216307
Values of k such that 10*k+3 and 10*k+9 alone are prime between 10*k and 10*k+9.
Original entry on oeis.org
2, 5, 8, 17, 23, 26, 35, 37, 38, 44, 50, 56, 59, 65, 73, 101, 110, 112, 122, 128, 143, 149, 154, 155, 161, 175, 197, 206, 233, 239, 254, 269, 290, 296, 308, 320, 331, 332, 353, 373, 392, 401, 407, 413, 425, 428, 464, 467, 479, 490, 499, 500, 511, 527, 530
Offset: 1
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t = {}; Do[ps = Select[Range[10*n, 10*n + 9], PrimeQ]; If[ps == {10*n + 3, 10*n + 9}, AppendTo[t, n]], {n, 0, 782}]; t (* T. D. Noe, Sep 04 2012 *)
Select[Range[800],Boole[PrimeQ[Range[10 #,10 #+9]]]=={0,0,0,1,0,0,0,0,0,1}&] (* Harvey P. Dale, Apr 23 2019 *)
A216308
Values of k such that 10*k+7 and 10*k+9 are the only primes between 10*k and 10*k+9.
Original entry on oeis.org
34, 127, 202, 223, 226, 265, 352, 355, 412, 433, 454, 463, 496, 619, 694, 730, 838, 853, 859, 967, 976, 1000, 1003, 1042, 1093, 1105, 1171, 1177, 1321, 1339, 1399, 1438, 1444, 1486, 1528, 1741, 1759, 1765, 1774, 1783, 1795, 1828, 1969, 2047, 2050, 2071, 2080
Offset: 1
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t = {}; Do[ps = Select[Range[10*n, 10*n + 9], PrimeQ]; If[ps == {10*n + 7, 10*n + 9}, AppendTo[t, n]], {n, 0, 2599}]; t (* T. D. Noe, Sep 04 2012 *)
A309871
Numbers n for which 18n+1, 18n+5, 18n+7, 18n+11, 18n+13 and 18n+17 are primes.
Original entry on oeis.org
892, 2432, 156817, 806697, 822937, 1377022, 1389412, 1418007, 1619642, 1753552, 2017732, 2058647, 2329302, 2554142, 2703347, 3058772, 3135107, 3326522, 3391797, 3723457, 4126867, 4132782, 4171422, 4411837, 4610252, 6378487, 6440087, 6878987, 6897782, 6991547
Offset: 1
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tot[n_] := Select[Range[n], CoprimeQ[#, n] &]; m = 18; t = tot[m]; aQ[n_] := AllTrue[m * n + t, PrimeQ]; Select[Range[10^6], aQ] (* Amiram Eldar, Aug 22 2019 *)
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x = 1
for i in range(5000000):
if (18*i+1 in Primes()
and 18*i+5 in Primes()
and 18*i+7 in Primes()
and 18*i+11 in Primes()
and 18*i+13 in Primes()
and 18*i+17 in Primes()):
print(str(x)+" "+str(i))
x += 1
A354589
Primes p starting a sequence of 4 consecutive primes whose final digits are 1,3,7,9 (in any order).
Original entry on oeis.org
11, 23, 47, 53, 67, 83, 89, 101, 109, 149, 167, 191, 193, 197, 199, 211, 251, 257, 263, 383, 443, 449, 461, 487, 557, 563, 587, 593, 599, 613, 647, 659, 739, 757, 761, 821, 859, 983, 991, 1061, 1063, 1069, 1117, 1217, 1223, 1283, 1301, 1303, 1367, 1433, 1439, 1447, 1481, 1553, 1567, 1571, 1579
Offset: 1
a(3) = 47 is in the sequence because the 4 consecutive primes starting with 47 are 47, 53, 59, 61, and their final digits 7,3,9,1 are a permutation of 1,3,7,9.
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P:= select(isprime, [seq(i,i=3..2000,2)]):
P1:= P mod 10:
P[select(i -> convert(P1[i..i+3],set) = {1,3,7,9}, [$1..nops(P)-3])];
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Select[Partition[Prime[Range[300]], 4, 1], Sort[Mod[#, 10]] == {1, 3, 7, 9} &][[;; , 1]] (* Amiram Eldar, Aug 19 2022 *)
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from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
p = [2, 3, 5, 7]
while True:
if set(map(lambda x: x%10, p)) == {1, 3, 7, 9}: yield p[0]
p = p[1:] + [nextprime(p[-1])]
print(list(islice(agen(), 60))) # Michael S. Branicky, Aug 18 2022
A064963
10000n+1, 10000n+3, 10000n+7, 10000n+9 are all primes.
Original entry on oeis.org
676, 1189, 2515, 2830, 8224, 9001, 10621, 10786, 17611, 18640, 20983, 21277, 23419, 28468, 31450, 37720, 41530, 41596, 42025, 45238, 47212, 49912, 50992, 52222, 53815, 65827, 70786, 77044, 82324, 88297, 88918, 96193, 99262, 101992
Offset: 1
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Select[Range[10^5], PrimeQ[10^4# + 1] && PrimeQ[10^4# + 3] && PrimeQ[10^4# + 7] && PrimeQ[10^4# + 9] &]
Select[Range[120000],AllTrue[10000#+{1,3,7,9},PrimeQ]&] (* Harvey P. Dale, Mar 18 2022 *)
A064968
Numbers k such that 1000000000k+1, 1000000000k+3, 1000000000k+7, 1000000000k+9 are all primes.
Original entry on oeis.org
14965, 16813, 20767, 23083, 34270, 40198, 93238, 112096, 189802, 192484, 251248, 333946, 334969, 363514, 374107, 375127, 376765, 383473, 405046, 419458, 462928, 498139, 649948, 703246, 704374, 732463, 767101, 781885, 806467, 812902, 842428
Offset: 1
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Select[Range[10^6], PrimeQ[10^9# + 1] && PrimeQ[10^9# + 3] && PrimeQ[10^9# + 7] && PrimeQ[10^9# + 9] &]
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{ n=0; for (m=1, 10^9, b=10^9*m; if(isprime(b + 1) && isprime(b + 3) && isprime(b + 7) && isprime(b + 9), write("b064968.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 01 2009
A088264
Smallest number k > 0 such that prefixing k to the n-th quadruple in the set {(1,3,7,9), (11,13,17,19), (21,23,27,29), ...} yields all primes.
Original entry on oeis.org
1, 189, 8, 94, 156, 32, 34, 18, 14, 1, 1653, 101, 2764, 99, 326, 715, 144, 1322, 4300, 768, 122, 67, 72, 500, 427, 3, 77, 22, 285, 119, 25, 294, 632, 55, 51, 3974, 217, 1230, 1022, 346, 1461, 260, 19, 9, 536, 463, 3, 299, 1, 69, 539, 1285, 1833, 116, 397, 3951
Offset: 1
a(2) = 189 as 189 is the smallest number such that 18911, 18913, 18917 and 18919 are all prime.
-
f:= proc(n) local R,k,p;
R:= map(`+`,[1,3,7,9],10*(n-1));
p:= 10^(ilog10(R[1])+1);
for k from 1 do
if map(t -> isprime(t+p*k), R) = [true,true,true,true] then return k fi
od
end proc:
map(f, [$1..60]); # Robert Israel, Jun 18 2017
A097639
a(n) is the smallest number m such that for the n-digit number s=10^(n-1)+ m, 10*s+1, 10*s+3, 10*s+7 and 10*s+9 are primes.
Original entry on oeis.org
0, 0, 48, 300, 111, 234, 1395, 546, 2526, 5742, 753, 12369, 5658, 94572, 6744, 134649, 32523, 43071, 213927, 256116, 8172, 431904, 57138, 433125, 123225, 711447, 318501, 40758, 150063, 184602, 134661, 377778, 129048, 504678, 88113, 3174738
Offset: 1
a(4)=300 because 10(10^3+300)+ 1, 10(10^3+300)+ 3, 10(10^3+300)+ 7 and 10(10^3+300)+1, are primes and 300 is the smallest number with this property.
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a[n_]:=(For[m=0, !(PrimeQ[10^n+10m+1]&&PrimeQ[10^n+10m+3]&&PrimeQ[ 10^n+10m+7]&&PrimeQ[10^n+10m+9]), m++ ];m);Table[a[n], {n, 43}]
Table[Module[{m=0,s=10^n},While[AnyTrue[10(s+m)+{1,3,7,9},CompositeQ],m++];m],{n,0,35}] (* Harvey P. Dale, Sep 19 2022 *)
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isok(m, n) = my(s=10^(n-1)+ m); ispseudoprime(10*s+1) && ispseudoprime(10*s+3) && ispseudoprime(10*s+7) && ispseudoprime(10*s+9);
a(n) = my(m=0); while (!isok(m, n), m++); m; \\ Michel Marcus, Aug 09 2023
A178082
Numbers k for which 5*k-4, 5*k-2, 5*k+2, and 5*k+4 are primes.
Original entry on oeis.org
3, 21, 39, 165, 297, 375, 417, 651, 693, 1131, 1887, 2601, 3129, 3147, 3213, 3609, 3783, 3885, 4203, 4455, 5061, 6345, 6969, 8757, 10269, 11067, 12597, 13443, 13899, 14445, 15453, 15939, 16209, 16545, 17763, 19569, 19827, 20223, 21969, 23307
Offset: 1
The associated prime quadruplets start as:
11, 13, 17, 19; (for n = 3)
101, 103, 107, 109; (for n = 21)
191, 193, 197, 199; (for n = 39)
821, 823, 827, 829;
1481, 1483, 1487, 1489;
1871, 1873, 1877, 1879;
2081, 2083, 2087, 2089;
3251, 3253, 3257, 3259;
3461, 3463, 3467, 3469;
5651, 5653, 5657, 5659;
9431, 9433, 9437, 9439;
13001, 13003, 13007, 13009;
15641, 15643, 15647, 15649;
15731, 15733, 15737, 15739;
16061, 16063, 16067, 16069;
18041, 18043, 18047, 18049;
18911, 18913, 18917, 18919;
19421, 19423, 19427, 19429.
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[n: n in [0..1000]| IsPrime(5*n - 4) and IsPrime(5*n - 2) and IsPrime(5*n + 2) and IsPrime(5*n + 4)]; // Vincenzo Librandi, Nov 30 2010
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Flatten[Table[If[PrimeQ[5*n + 2] && PrimeQ[5*n - 2] && PrimeQ[5*n + 4] && PrimeQ[5*n - 4], n, {}], {n, 0, 10000}]]
Select[Range[25000],AllTrue[5#+{4,2,-2,-4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 03 2018 *)
A216309
The prime ending in 1 is the only prime in a decade.
Original entry on oeis.org
181, 211, 241, 421, 631, 661, 691, 811, 1021, 1051, 1171, 1201, 1381, 1471, 1511, 1531, 1801, 1811, 1831, 1951, 2161, 2221, 2251, 2311, 2521, 2621, 2731, 2861, 2971, 3001, 3121, 3191, 3271, 3331, 3361, 3391, 3571, 3931, 4111, 4201, 4231, 4261, 4621, 4691
Offset: 1
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t = {}; Do[ps = Select[Range[10*n, 10*n + 9], PrimeQ]; If[ps == {10*n + 1}, AppendTo[t, ps[[1]]]], {n, 0, 588}]; t (* T. D. Noe, Sep 04 2012 *)
Select[10*Range[500]+1,PrimeQ[#]&&AllTrue[#+{2,6,8},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 01 2018 *)
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