OFFSET
1,1
COMMENTS
The sequence has repetitions since different p's and q's will give the same perfect power. Remove the andmap in the program if you want the repetitions.
Includes all perfect powers, pp, (A001597) congruent +/- 1 (modulo 6). Also if pp-9 or pp-12 is a prime or if (pp -2)/3 or (pp-3)/3 is a prime.
The number of perfect powers of the form 3p + q + 3 <= 10^n: 0,5,21,56,157,433,...,. - Robert G. Wilson v, Jun 21 2006
In the first one million integers there are 1111 perfect powers (A070428) of which only 433 of them are of the form 3p + q + 3.
FORMULA
a(n)=3p+q+3 where p and q are primes and a(n) is a perfect power.
EXAMPLE
a(5)=49 since 3*3+37+3=49 = 5*3+31+3 = 3*11+13+3 = 3*13+7+7 = 7^2.
6859 = 19^3 is in the sequence because there are 116 different ways to combine primes of the form 3p + q + 3, beginning with p=5 & q=6841 and ending with p=2281 & q=13.
MAPLE
with(numtheory); egcd := proc(n) local L; L:=map(proc(z) z[2] end, ifactors(n)[2]); igcd(op(L)) end: PW:=[]: for z to 1 do for j from 1 to 100 do for k from 1 to 100 do p:=ithprime(j); q:=ithprime(k); x:=3*p+q+3; if egcd(x)>1 and andmap(proc(w) not(w[3]=x) end, PW) then PW:=[op(PW), [p, q, x]] fi od od od; PW; map(proc(z) z[3] end, PW);
MATHEMATICA
fQ[n_] := GCD @@ Last /@ FactorInteger@n > 1; lst = {}; Do[p = Prime@j; q = Prime@k; x = 3p + q + 3; If[fQ@x, AppendTo[lst, x]], {j, 340}, {k, PrimePi[6856 - 3Prime@j]}]; Union@lst (* Robert G. Wilson v *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Kehowski, Oct 05 2005
EXTENSIONS
Edited, corrected and extended by Robert G. Wilson v, Jun 21 2006
STATUS
approved