A285438 - OEIS

%I #28 Jan 29 2020 02:52:11

%S 4,8,9,16,32,36,64,128,144,256,324,512,576,1024,2048,2304,2744,2916,

%T 4096,8192,9216,16384,26244,32768,36864,65536,131072,147456,236196,

%U 262144,524288,589824,941192,1048576,2097152,2125764,2359296,4194304,8388608,9437184

%N Perfect powers that are also the sum of two powers of a prime p.

%C Integers n such that there exist integers i, j, k, m, p with i, j >= 0, m, k >= 2 and p prime, such that n = m^k = p^i + p^j.

%C These are numbers of the form 2^r = 2^(r-1) + 2^(r-1) when r >= 2, numbers of the form (3*2^r)^2 = 2^(2*r) + 2^(2*r+3) and numbers of the form (2*p^r)^k = p^(r*k) + p^(r*k+1) when p = 2^k - 1 is a Mersenne prime. [Edited by _Jinyuan Wang_, Nov 30 2019]

%C If n = p^i + p^j is a term with exactly two sets of integer solutions (p, i, j), where i <= j, then n must be 36 = 6^2 = 2^2 + 2^5 = 3^2 + 3^3 or of the form 2^k = 2^(k-1) + 2^(k-1) = p^0 + p^1 where p = 2^k - 1 is a Mersenne prime. There is no n = p^i + p^j in this sequence with at least three sets of integer solutions (p, i, j), where i <= j. - _Jinyuan Wang_, Nov 30 2019

%H Robert Israel, <a href="/A285438/b285438.txt">Table of n, a(n) for n = 1..6655</a>

%H W. Weakley, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.9.941">Problem 11936</a>, Amer. Math. Monthly, 123 (2016), 941.

%e 324 = 18^2 = 3^4 + 3^5.

%p N:= 10^9: # to get all terms <= N

%p R1:= {seq(2^i,i=2..ilog2(N))}:

%p R2:= {seq(9*2^(2*r), r=0..ilog2(floor(N/9))/2)}:

%p R3:= {seq(seq(2^k*(2^k-1)^(r*k),r=1..floor(log[2^k-1](N/2^k)/k)),k=select(t -> isprime(2^t-1),[$2..ilog2(N)]))}:

%p sort(convert(R1 union R2 union R3, list)); # _Robert Israel_, Apr 25 2017

%o (PARI) upto(nn) = {my(v=List([]), k=1); for(r=2, logint(nn, 2), listput(v, 2^r)); for(r=0, logint(nn\9, 4), listput(v, 9*4^r)); while((2*2^k-2)^k<nn, k=nextprime(k+1); if(isprime(2^k-1), for(r=1, logint(nn\2^k, q=(2^k-1)^k), listput(v, 2^k*q^r)))); Set(v); }

%o upto(10^9) \\ _Jinyuan Wang_, Nov 30 2019

%Y Cf. A000668, A048645, A072103, A161792, A282550.

%K nonn

%O 1,1

%A _Michael Josephy_, Apr 18 2017

%E a(19)-a(40) from _Robert Israel_, Apr 25 2017