A157468
Primes of the form sqrt(p-1)-1, where p is a prime.
Original entry on oeis.org
3, 5, 13, 19, 23, 53, 73, 83, 89, 109, 149, 179, 223, 229, 239, 263, 269, 283, 313, 349, 383, 419, 439, 443, 463, 569, 593, 643, 653, 673, 739, 859, 863, 919, 929, 1009, 1069, 1093, 1123, 1289, 1319, 1373, 1409, 1429, 1433, 1439, 1459
Offset: 1
3 is in the sequence because 3 = sqrt(17 - 1) - 1, where 17 is prime.
5 is in the sequence because 5 = sqrt(37 - 1) - 1, where 37 is prime.
A238048
Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k is the increasing list of all primes p such that (p+k)^2+k is also prime.
Original entry on oeis.org
3, 7, 5, 5, 13, 13, 3, 7, 19, 19, 7, 11, 11, 31, 23, 5, 31, 13, 19, 37, 53, 3, 13, 43, 23, 47, 43, 73, 7, 5, 19, 67, 29, 59, 79, 83, 11, 13, 11, 29, 73, 31, 61, 97, 89, 3, 23, 43, 19, 59, 109, 41, 67, 103, 109, 13, 17, 29, 73, 23, 73, 157, 43, 71, 109, 149
Offset: 1
Column k=3 contains prime 47 because (47+3)^2+3 = 2503 is prime.
Square array A(n,k) begins:
3, 7, 5, 3, 7, 5, 3, 7, ...
5, 13, 7, 11, 31, 13, 5, 13, ...
13, 19, 11, 13, 43, 19, 11, 43, ...
19, 31, 19, 23, 67, 29, 19, 73, ...
23, 37, 47, 29, 73, 59, 23, 79, ...
53, 43, 59, 31, 109, 73, 29, 103, ...
73, 79, 61, 41, 157, 83, 31, 109, ...
83, 97, 67, 43, 163, 103, 41, 127, ...
-
A:= proc(n, k) option remember; local p;
p:= `if`(n=1, 1, A(n-1, k));
do p:= nextprime(p);
if isprime((p+k)^2+k) then return p fi
od
end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..11);
-
A[n_, k_] := A[n, k] = Module[{p}, For[p = If[n == 1, 1, A[n-1, k]] // NextPrime, True, p = NextPrime[p], If[PrimeQ[(p+k)^2+k], Return[p]]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 11}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)
A238664
Primes p such that (p+2)^2+2 is prime but (p+1)^2+1 is not prime.
Original entry on oeis.org
7, 31, 37, 43, 79, 97, 103, 241, 271, 307, 367, 373, 421, 499, 547, 571, 601, 607, 709, 751, 883, 907, 967, 1033, 1129, 1213, 1231, 1237, 1327, 1423, 1597, 1609, 1621, 1747, 1801, 1867, 1933, 1951, 1993, 2017, 2131, 2137, 2203, 2221, 2281, 2287, 2647, 2659
Offset: 1
A238665
Primes p such that (p+3)^2+3 is prime but (p+j)^2+j is not prime for all 0
Original entry on oeis.org
11, 47, 59, 61, 67, 71, 127, 131, 137, 151, 157, 227, 337, 347, 353, 431, 467, 509, 521, 557, 577, 599, 613, 617, 619, 631, 683, 691, 701, 733, 743, 773, 857, 911, 983, 997, 1013, 1039, 1051, 1097, 1151, 1153, 1193, 1201, 1307, 1321, 1453, 1471, 1531, 1607
Offset: 1
A238666
Primes p such that (p+4)^2+4 is prime but (p+j)^2+j is not prime for all 0
Original entry on oeis.org
29, 41, 113, 163, 173, 199, 211, 251, 449, 479, 491, 503, 659, 661, 809, 823, 941, 1031, 1171, 1181, 1259, 1361, 1669, 1753, 1759, 1861, 1879, 1901, 1999, 2039, 2081, 2141, 2161, 2213, 2273, 2371, 2473, 2539, 2579, 2591, 2633, 2819, 2903, 2939, 2969, 3011
Offset: 1
A238667
Primes p such that (p+5)^2+5 is prime but (p+j)^2+j is not prime for all 0
Original entry on oeis.org
193, 331, 409, 457, 487, 787, 829, 991, 1087, 1117, 1249, 1297, 1303, 1543, 1627, 2251, 2311, 2377, 2521, 2767, 2857, 3061, 3067, 3739, 3769, 3907, 3931, 4027, 4057, 4099, 4159, 4567, 5023, 5281, 5407, 5581, 5749, 5827, 5839, 6073, 6379, 7039, 7879, 7963, 8017
Offset: 1
A238668
Primes p such that (p+6)^2+6 is prime but (p+j)^2+j is not prime for all 0
Original entry on oeis.org
139, 523, 563, 769, 853, 1019, 1489, 1553, 1559, 1583, 1693, 1723, 1949, 2239, 2339, 2393, 2423, 3469, 3779, 3863, 4073, 4133, 4273, 4283, 4483, 4663, 4969, 5233, 5503, 5683, 5869, 5953, 6269, 6299, 6473, 6569, 6959, 7229, 7309, 8233, 8513, 8573, 8839, 9749
Offset: 1
-
pnpQ[n_]:=Module[{c=Table[(n+j)^2+j,{j,6}]},NoneTrue[Most[c], PrimeQ] &&PrimeQ[Last[c]]]; Select[Prime[Range[1500]],pnpQ] (* This program uses the function NoneTrue from Mathematica version 10 *) (* Harvey P. Dale, Jul 26 2014 *)
A238669
Primes p such that (p+7)^2+7 is prime but (p+j)^2+j is not prime for all 0
Original entry on oeis.org
107, 293, 359, 389, 397, 401, 433, 461, 647, 727, 797, 821, 977, 1063, 1163, 1229, 1301, 1367, 1427, 1451, 1499, 1571, 1657, 1721, 1987, 2099, 2111, 2179, 2207, 2351, 2447, 2707, 2797, 2801, 2861, 2957, 3037, 3187, 3221, 3457, 3463, 3527, 3541, 3557, 3607
Offset: 1
-
Select[Prime[Range[600]],PrimeQ[(#+7)^2+7]&&NoneTrue[Table[(#+j)^2+j,{j,6}],PrimeQ]&] (* Harvey P. Dale, Apr 04 2024 *)
A238670
Primes p such that (p+8)^2+8 is prime but (p+j)^2+j is not prime for all 0
Original entry on oeis.org
181, 277, 541, 937, 1381, 1741, 2551, 2617, 2677, 3433, 3919, 4231, 4657, 4933, 5923, 6337, 6481, 6781, 7669, 7717, 7867, 8161, 8167, 8287, 8329, 8389, 8647, 8707, 9013, 9151, 9397, 9661, 9739, 9967, 10651, 11059, 11287, 11743, 11887, 12421, 12457, 12697
Offset: 1
-
Select[Prime[Range[1600]],PrimeQ[Table[(#+n)^2+n,{n,8}]]=={False, False, False, False, False,False,False,True}&] (* Harvey P. Dale, Dec 17 2016 *)
A238671
Primes p such that (p+9)^2+9 is prime but (p+j)^2+j is not prime for all 0
Original entry on oeis.org
101, 191, 233, 311, 881, 1103, 1291, 1733, 1831, 1931, 2011, 2029, 2113, 2129, 2269, 2543, 2843, 3023, 3089, 3163, 3299, 3491, 3701, 3761, 3943, 4051, 4391, 4583, 4951, 5333, 5441, 5743, 5801, 6211, 6421, 6491, 7019, 7069, 7121, 7253, 7331, 8081, 8171, 8293
Offset: 1
-
Select[Prime[Range[1100]],PrimeQ[(#+9)^2+9]&&NoneTrue[Table[(#+j)^2+j,{j,8}],PrimeQ]&] (* Harvey P. Dale, Jul 02 2022 *)
Showing 1-10 of 22 results.
Comments