Original entry on oeis.org
3373, 753569, 2146687, 3048623, 6539201, 8120599, 10218311, 17373977, 18609623, 19034161, 32461757, 44738873, 59776469, 69426529, 72511711, 77854481, 88121123, 116930167, 133432829, 299418307, 338608871, 413493623, 458314009, 679151437
Offset: 1
3373 + 2 = 3375 = 3^3*5^3. 753569 + 1 = 753571 = 7^3*13^3.
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N:= 10^10: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..floor((N+2)^(1/3)/3))]):
R:= NULL:
for i from 1 to nops(P) do
for j from 1 to i-1 do
p:= (P[i]*P[j])^3-2;
if p > N then break fi;
if isprime(p) then R:= R, p fi
od od:
sort([R]); # Robert Israel, Jun 05 2018
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f3[n_]:=FactorInteger[n][[1,2]]==3&&Length[FactorInteger[n]]==2&&FactorInteger[n][[2, 2]]==3; lst={};Do[p=Prime[n];If[f3[p+2],AppendTo[lst,p]],{n,4,4*9!}]; lst
csfsQ[n_]:=Module[{c=Surd[n+2,3]},SquareFreeQ[c]&&PrimeOmega[c]==2]; Select[Prime[Range[353*10^5]],csfsQ] (* Harvey P. Dale, Jan 07 2018 *)
A216981
Primes of the form n^7+2.
Original entry on oeis.org
2, 3, 4782971, 1801088543, 1174711139839, 3938980639169, 93206534790701, 425927596977749, 1107984764452583, 2149422977421877, 7416552901015627, 19891027786401119, 307732862434921877, 830512886046548069, 1042842864990234377, 3678954248903875651
Offset: 1
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[a: n in [0..500] | IsPrime(a) where a is n^7+2]; // Vincenzo Librandi, Mar 15 2013
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lst={}; Do[p=n^7+2; If[PrimeQ[p], AppendTo[lst, p]], {n, 6!}]; lst
Select[Table[n^7 + 2, {n, 0, 400}], PrimeQ] (* Vincenzo Librandi, Mar 15 2013 *)
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v=select(n->isprime(n^7+2),vector(2000,n,n-1)); /* A216980 */
vector(#v, n, v[n]^7+2)
/* Joerg Arndt, Sep 21 2012 */
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select(isprime, vector(2000,n,(n-1)^7+2)) \\ Charles R Greathouse IV, Sep 21 2012
A259189
Semiprimes of the form n^3 + 2.
Original entry on oeis.org
10, 218, 514, 731, 1333, 2199, 2746, 3377, 4915, 5834, 6861, 8002, 9263, 12169, 15627, 29793, 35939, 42877, 54874, 59321, 68923, 117651, 125002, 132653, 148879, 185195, 205381, 314434, 405226, 421877, 474554, 531443, 592706, 658505, 704971
Offset: 1
Cf.
A237040 (similar sequence with n^3+1).
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IsSP:=func;[r:n in [1..1000]|IsSP(r) where r is 2+n^3];
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Select[Range[100]^3 + 2, PrimeOmega[#] == 2 &] (* Alonso del Arte, Jun 20 2015 *)
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is(n)=bigomega(n^3 + 2)==2 \\ Anders Hellström, Sep 07 2015
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use ntheory ":all"; my @sp = grep { scalar(factor($))==2 } map { $**3+2 } 1..100; say "@sp"; # Dana Jacobsen, Sep 07 2015
A283698
Numbers k such that {k^2 + 2, k^2 + 4} and {k^3 + 2, k^3 + 4} are twin prime pairs.
Original entry on oeis.org
1, 3, 45, 2055, 39033, 48585, 101535, 104553, 112383, 117723, 129315, 152553, 170793, 178095, 234483, 246435, 258093, 272403, 304845, 306885, 365343, 372663, 375813, 405393, 405975, 436425, 456903, 494193, 538965, 551475, 559713, 569805, 570033, 767895, 792903
Offset: 1
a(2) = 3, {3^2 + 2 = 11, 3^2 + 4 = 13 } and {3^3 + 2 = 29, 3^3 + 4 = 31} are twin prime pairs.
a(3) = 45, {45^2 + 2 = 2027, 45^2 + 4 = 2029 } and {45^3 + 2 = 91127, 45^3 + 4 = 91129} are twin prime pairs.
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Select[Range[1000000], PrimeQ[#^2 + 2] && PrimeQ[#^2 + 4] && PrimeQ[#^3 + 2] && PrimeQ[#^3 + 4] &]
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for(n=1, 100000, if(isprime(n^2+2) && isprime(n^2+4) && isprime(n^3+2) && isprime(n^3+4), print1(n, ", ")))
A284058
Numbers k such that {k + 2, k + 4} and {k^3 + 2, k^3 + 4} are twin prime pairs.
Original entry on oeis.org
1, 3, 69, 1719, 3555, 8535, 8625, 9765, 10065, 17955, 27939, 32319, 34209, 35445, 39159, 44769, 47415, 55329, 56235, 75615, 85929, 91965, 96219, 97545, 98895, 122385, 122595, 138075, 142695, 143649, 145719, 152025, 191829, 192975, 197955, 200379, 201819, 202059
Offset: 1
a(2) = 3, {3 + 2 = 5, 3 + 4 = 7} and {3^3 + 2 = 29, 3^3 + 4 = 31} are twin prime pairs.
a(3) = 69, {69 + 2 = 71, 69 + 4 = 73} and {69^3 + 2 = 328511, 69^3 + 4 = 328513} are twin prime pairs.
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Select[Range[1000000], PrimeQ[# + 2] && PrimeQ[# + 4] && PrimeQ[#^3 + 2] && PrimeQ[#^3 + 4] &]
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for(n=1, 100000,2; if(isprime(n+2) && isprime(n+4) && isprime(n^3+2) && isprime(n^3+4), print1(n, ", ")))
A214001
Numbers n such that n^2+2, n^3+2, n^4+2 and n^5+2 are all prime.
Original entry on oeis.org
0, 1, 909, 2055, 11925, 145881, 191079, 254199, 358875, 490215, 614241, 642105, 648261, 689655, 864159, 959595, 1030911, 1047585, 1056981, 1150335, 1366971, 1406571, 1669845, 1746525, 2299485, 2357751, 2491809, 2494329, 2629869, 2876859, 3162159, 3220041, 3257595
Offset: 1
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Select[Range[3500000], And@@PrimeQ/@(Table[n^i+2, {i, 2, 5}]/.n->#)&]
Select[Range[0,33*10^5],AllTrue[#^Range[2,5]+2,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 21 2018 *)
Comments