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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

Table of n, a(n) for n=1..38.

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

Subsequence of A001597, A000961.

Cf. A000079, A001248, A108605, A109611.

Sequence in context: A114130 A130100 A226230 * A173336 A277925 A173745

Adjacent sequences:  A258397 A258398 A258399 * A258401 A258402 A258403

KEYWORD

nonn

AUTHOR

Carlos Eduardo Olivieri, May 28 2015

EXTENSIONS

a(28)-a(38) from Giovanni Resta, Jun 23 2015

STATUS

approved

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Last modified May 6 17:00 EDT 2021. Contains 343586 sequences. (Running on oeis4.)