A274525
Prime numbers p such that p - 2, p^2 - p - 1, p^2 - p + 1 are prime numbers.
Original entry on oeis.org
7, 139, 1789, 2731, 4159, 5641, 13339, 13399, 19429, 21739, 22369, 32059, 32911, 33601, 42571, 45319, 54541, 55339, 65449, 68821, 106189, 108499, 111871, 132859, 136399, 138079, 141511, 142981, 148201, 149629, 152041, 152839, 173431, 174049, 178249
Offset: 1
5 - 2 = 3 prime, 5 prime, 5*(5-1) - 1 = 19 prime, 5*(5-1) + 1 = 21 composite, so 5 is not in the sequence.
7 - 2 = 5 prime, 7 prime, 7*(7-1) - 1 = 41 prime, 7*(7-1) + 1 = 43 prime so 7 is in the sequence.
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Select[Prime[Range[100]], Union[PrimeQ[{# - 2, #^2 - # - 1, #^2 - # + 1}]] == {True} &] (* Alonso del Arte, Jun 27 2016 *)
Select[Prime[Range[17000]],AllTrue[{#-2,#^2-#-1,#^2-#+1},PrimeQ]&] (* Harvey P. Dale, Jun 20 2024 *)
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lista(nn) = forprime(p=2, nn, if (isprime(p-2) && isprime(p^2-p-1) && isprime(p^2-p+1), print1(p, ", "))); \\ Michel Marcus, Jul 07 2016
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from sympy import isprime, primerange
def aupto(n):
t = []
for p in primerange(2, n+1):
if isprime(p-2) and isprime(p**2 - p - 1) and isprime(p**2 - p + 1):
t.append(p)
return t # Paul Muljadi, Jun 21 2024
A274527
Prime numbers p such that p+2, p^2+p-1, p^2+p+1, p^2+3*p+1, p^2+3*p+3 are all prime numbers.
Original entry on oeis.org
5, 2729, 26449079, 27188279, 44521679, 46090379, 52736249, 62320439, 70777979, 92520539, 109505969, 192153149, 274448789, 288269519, 343801919, 359240069, 515694899, 521594639, 527159429, 660223409, 809600819, 857353139, 921868289, 945420629, 1000777049
Offset: 1
2 first prime, 2+2=4 composite.
3 prime, 3+2=5 prime, 3^2+3-1=11 prime, 3^2+3+1=13 prime, 3^2+3*3+1=19 prime, 3^2+3*3+3=21 composite.
5 prime, 5+2=7 prime, 5^2+5-1=29 prime, 5^2+5+1=31 prime, 5^2+3*5+1=41 prime, 5^2+3*5+3=43 prime so a(1)=5
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Select[Prime[Range[50890000]],AllTrue[{#+2,#^2+#-1,#^2+#+1,#^2+3#+1,#^2+3#+3},PrimeQ]&] (* Harvey P. Dale, Aug 13 2021 *)
A228978
Primes p such that p and p+2, p^2+p-1 and p^2+p+1, and (p^2+p-1)^2+p^2+p-2 and (p^2+p-1)^2+p^2+p are three pairs of twin primes.
Original entry on oeis.org
11767181, 35057849, 84428051, 91460249, 105929711, 115401719, 162790781, 197352401, 217761851, 235863209, 266250839, 284597741, 370000511, 386278019, 554761451, 576412271, 581549669, 592975109, 611599661, 625806761, 626450411, 655727771, 670280591, 680468669, 744737111, 883687349, 1085880641, 1119813311, 1139369111
Offset: 1
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tptpQ[n_]:=Module[{p2=n^2+n},AllTrue[{p2-1,p2+1,(p2-1)^2+p2-2,(p2-1)^2+ p2},PrimeQ]]; Select[Transpose[Select[Partition[ Prime[ Range[ 58*10^6]],2,1], #[[2]]-#[[1]]==2&]][[1]],tptpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 10 2014 *)
Showing 1-3 of 3 results.
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