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Primes of the form k^2 + prime(k).
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3, 7, 23, 83, 181, 239, 563, 1013, 1447, 1607, 2129, 2729, 3167, 3881, 4673, 5849, 6481, 7489, 8563, 9719, 11813, 18713, 21563, 25247, 27197, 29221, 33469, 36467, 47977, 50683, 51599, 56237, 69257, 71389, 75731, 96893, 107119, 115163
EXAMPLE
3167 is here because 54^2 + prime(54) = 54^2 + 251 = 3167, which is prime.
MATHEMATICA
Select[Table[k^2 + Prime[k], {k, 1000}], PrimeQ] (* Harvey P. Dale, Feb 16 2011 *)
PROG
(Magma) [ a: k in [0..10000] | IsPrime(a) where a is k^2 + NthPrime(k) ]; // Vincenzo Librandi, Apr 14 2011
Primes of the form k^2 - prime(k).
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23, 71, 107, 263, 487, 677, 787, 1427, 1583, 2081, 3319, 5393, 8713, 10247, 11071, 12377, 18257, 20477, 24659, 26573, 29243, 29927, 33487, 34949, 37223, 37991, 41981, 51449, 60917, 64937, 66977, 71167, 83357, 85667, 99013, 100271, 109313, 110629, 118757
COMMENTS
The two primes prime(k) and k^2-prime(k) are a Goldbach partition of k^2. - T. D. Noe, Apr 14 2011
EXAMPLE
23 is here because 6^2 - prime(6) = 36 - 13 = 23.
MATHEMATICA
Select[Table[k^2 - Prime[k], {k, 1000}], PrimeQ] (* T. D. Noe, Apr 14 2011 *)
PROG
(Magma) [ a: k in [0..10000] | IsPrime(a) where a is k^2-NthPrime(k) ]; // Vincenzo Librandi, Apr 14 2011
Numbers k such that k^2 + prime(k) and k^2 - prime(k) are both primes.
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12, 30, 60, 96, 336, 660, 702, 756, 984, 990, 1188, 1302, 1488, 1830, 1866, 2070, 2142, 2340, 2586, 2874, 2910, 3618, 3714, 3750, 3774, 3906, 4008, 4470, 4512, 4902, 5094, 5754, 6012, 6174, 6432, 6840, 6846, 6930, 7446, 7578, 7734, 8064, 8190, 8328
EXAMPLE
12 is in the sequence because 144 +/- 37 = 181 and 107 which are both primes.
k=30 is a term: 30^2 = 900, prime(30) = 113, 900+113 = 1013 and 900-113 = 787, both primes.
MATHEMATICA
Select[ Range[10^4], PrimeQ[ #^2 + Prime[ # ]] && PrimeQ[ #^2 - Prime[ # ]] &]
PROG
(PARI) for(n=1, 20000, if(isprime(n^2+prime(n)) && isprime(n^2-prime(n)), print1(n, " ")))
(PARI) { n=0; default(primelimit, 6100000); for (m=1, 10^9, if (isprime(m^2 + prime(m)) && isprime(m^2 - prime(m)), write("b064483.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 16 2009
Numbers n such that n^2 + pi(n) is prime.
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2, 3, 7, 12, 18, 21, 36, 37, 42, 45, 52, 55, 60, 61, 65, 68, 70, 79, 84, 95, 98, 113, 130, 135, 143, 145, 155, 180, 181, 185, 195, 205, 216, 222, 231, 239, 253, 262, 273, 275, 325, 332, 334, 354, 368, 370, 385, 402, 417, 421, 432, 433, 454, 462, 488, 505, 516
COMMENTS
Conjecture: the sequence is infinite.
EXAMPLE
a(6) = 21 : n^2+pi(n ) = 21^2 + pi(21) = 441+8 = 449 which is a prime.
MAPLE
with(numtheory): KD:= proc() local a; a:= n^2+pi(n); if isprime(a) then RETURN(n): fi; end: seq(KD(), n=1..2000);
MATHEMATICA
Select[Range[600], PrimeQ[#^2+PrimePi[#]]&] (* Harvey P. Dale, Jul 04 2018 *)
PROG
(PARI) v=List(); p=0; for(n=2, 1e4, p+=isprime(n); if(isprime(n^2+p), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Sep 04 2013
CROSSREFS
Cf. A077510 (numbers n such that n + pi(n) is a prime).
Prime(n) is included iff prime(n) + n^2 is also prime.
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2, 3, 7, 19, 37, 43, 79, 113, 151, 163, 193, 229, 251, 281, 317, 373, 397, 433, 463, 503, 577, 757, 827, 911, 953, 997, 1069, 1123, 1321, 1399, 1423, 1481, 1657, 1693, 1747, 2029, 2143, 2267, 2311, 2473, 2503, 2551, 2593, 2687, 2753, 2791, 2917, 3313, 3323
EXAMPLE
7 is in this sequence because 7+16=23.
19 is in this sequence because 19+64=83.
MATHEMATICA
Prime[Select[Range[500], PrimeQ[Prime[#] + #^2] &]]
PROG
(Magma) [NthPrime(n): n in [1..500] | IsPrime(NthPrime(n)+n^2)];
(PARI) lista(nn) = forprime (n=2, nn, if (isprime(n+primepi(n)^2), print1(n, ", "))); \\ Michel Marcus, Feb 04 2015
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