

A111026


Perfect powers (A001597) of the form 3p + q + 3, p & q are primes.


1



16, 25, 27, 32, 49, 121, 125, 128, 169, 225, 243, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1000, 1225, 1331, 1369, 1681, 1849, 2025, 2048, 2187, 2197, 2209, 2401, 2809, 3025, 3125, 3375, 3481, 3721, 3969, 4225, 4489, 4913, 5041, 5329, 5625, 5929, 6241
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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 pp9 or pp12 is a prime or if (pp 2)/3 or (pp3)/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.


LINKS

Table of n, a(n) for n=1..47.


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

Cf. A000040, A001597.
Sequence in context: A071524 A227651 A095409 * A124186 A274240 A176512
Adjacent sequences: A111023 A111024 A111025 * A111027 A111028 A111029


KEYWORD

nonn


AUTHOR

Walter Kehowski, Oct 05 2005


EXTENSIONS

Edited, corrected and extended by Robert G. Wilson v, Jun 21 2006


STATUS

approved



