|
| |
|
|
A258400
|
|
Perfect powers m^k such that m, k and m+k are primes.
|
|
0
|
|
|
|
8, 9, 25, 32, 121, 289, 841, 1681, 2048, 3481, 5041, 10201, 11449, 18769, 22201, 32041, 36481, 38809, 51529, 57121, 72361, 78961, 96721, 120409, 131072, 175561, 185761, 212521, 271441, 323761, 358801, 380689, 410881, 434281, 654481, 674041, 683929, 734449
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Necessarily either m or k = 2, thus if a(n) is even, it is a power of 2 with odd prime exponent, otherwise (if a(n) is odd), it is a square of odd prime.
For each term m^k, there will be another k^m.
a(3), a(5), a(11) are of the form n! + 1.
Let F(m,k) = m*k, such that m^k = a(n), so A108605 is a subsequence of F. For example a(1) = 2^3 and F(2,3) = A108605(1).
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
a(1) = 8, because 8 = 2^3 and 2+3 = 5.
a(4) = 32, because 32 = 2^5 and 2+5 = 7.
a(5) = 121, because 121 = 11^2 and 11+2 = 13.
a(25) = 131072, because 131072 = 2^17 and 2+17 = 19.
|
|
|
MATHEMATICA
|
SmallestDivisor[n_] := If[n == 1, 1, Divisors[n][[2]]]; perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; ppl = Select[Range[200000], perfectPowerQ]; base[n_] := ppl[[n]]^(1/exp[n]); exp[n_] := SmallestDivisor[GCD @@ FactorInteger[ppl[[n]]][[All, 2]] ]; pp2l = Table[ {base[n], exp[n]}, {n, Length[ppl]}]; p[n_] := pp2l[[n]][[1]]; q[n_] := pp2l[[n]][[2]]; lt = Select[Range[Length[pp2l]], PrimeQ[p[#]] && PrimeQ[q[#]] && PrimeQ[p[#] + q[#]] &]; ppl[[lt]]
Select[Range[10^6], Length[f = FactorInteger@ #] == 1 && PrimeQ@ f[[1, 2]] && PrimeQ@ Total@ f[[1]] &] (* Giovanni Resta, Jun 23 2015 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
