A048636
Primes of the form prime^3 + 2.
Original entry on oeis.org
29, 127, 24391, 357913, 571789, 1442899, 5177719, 18191449, 30080233, 73560061, 80062993, 118370773, 127263529, 131872231, 318611989, 344472103, 440711083, 461889919, 590589721, 756058033, 865523179, 1095912793, 1298596573, 1341919729, 1524845953, 1697936059
Offset: 1
a(2) = 127 = 5^3 + 2 and 5 is prime.
Cf.
A092402 (primes of the form p + 8),
A321891 (union of the two);
A188764 (primes of the form (product of distinct primes^3) + 2).
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select(isprime, [ithprime(i)^3+2$i=1..300])[]; # Alois P. Heinz, Jan 13 2025
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lst={};Do[s=Prime[n]^3;If[PrimeQ[p=s+2], AppendTo[lst, p]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 26 2008 *)
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forprime (p=2,1100,if(isprime(p^3+2),print1(p^3+2,", "))) \\ Hugo Pfoertner, Oct 30 2018
A089195
Primes p such that all prime factors of p-1 have exponent 2.
Original entry on oeis.org
2, 5, 37, 101, 197, 677, 4357, 5477, 8837, 12101, 16901, 17957, 21317, 28901, 42437, 44101, 52901, 98597, 106277, 148997, 164837, 184901, 217157, 220901, 224677, 324901, 401957, 417317, 427717, 454277, 476101, 509797, 682277, 792101, 820837
Offset: 1
101 is included because 100 = 2^2*5^2 only square factors. 109 is not because while 108=2^2*3^3 has a square only factor it also has a cube factor.
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Prepend[Select[Table[Prime[n],{n,70000}],Length[Union[Last/@FactorInteger[#-1]]]==1&&Union[Last/@FactorInteger[#-1]]=={2}&], 2] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)
seq[lim_] := Select[Select[Range[Floor[Surd[lim-1, 2]]], SquareFreeQ]^2 + 1, PrimeQ]; seq[10^6] (* Amiram Eldar, Jan 18 2025 *)
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list(lim) = select(isprime, apply(x -> x^2 + 1, select(issquarefree, vector(sqrtnint(lim-1, 2), i, i)))); \\ Amiram Eldar, Jan 18 2025
A188717
Primes p such that all prime factors of p-1 have exponent 4.
Original entry on oeis.org
2, 17, 1297, 1336337, 4477457, 29986577, 45212177, 126247697, 193877777, 406586897, 562448657, 916636177, 1416468497, 1944810001, 3208542737, 4162314257, 5006411537, 5972816657, 12444741137, 19565295377, 34188010001, 38167092497, 47156728337, 59553569297, 61505984017
Offset: 1
17-1 = 2^4, 1297-1 = 2^4*3^4, 1336337-1 = 2^4*17^4, 4477457-1 = 2^4*23^4, ...
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Prepend[Select[Table[Prime[n],{n,600000}],Length[Union[Last/@FactorInteger[#-1]]]==1&&Union[Last/@FactorInteger[#-1]]=={4}&], 2]
seq[lim_] := Select[Select[Range[Floor[Surd[lim-1, 2]]], SquareFreeQ]^4 + 1, PrimeQ]; seq[10^6] (* Amiram Eldar, Jan 18 2025 *)
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list(lim) = select(isprime, apply(x -> x^4 + 1, select(issquarefree, vector(sqrtnint(lim-1, 4), i, i)))); \\ Amiram Eldar, Jan 18 2025
a(1) = 2 inserted and a(23)-a(25) added by
Amiram Eldar, Jan 18 2025
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