A285438 Perfect powers that are also the sum of two powers of a prime p.

4, 8, 9, 16, 32, 36, 64, 128, 144, 256, 324, 512, 576, 1024, 2048, 2304, 2744, 2916, 4096, 8192, 9216, 16384, 26244, 32768, 36864, 65536, 131072, 147456, 236196, 262144, 524288, 589824, 941192, 1048576, 2097152, 2125764, 2359296, 4194304, 8388608, 9437184

Offset: 1

Author

Michael Josephy, Apr 18 2017

Keywords

• nonn

Comments

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.

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]

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..6655
• W. Weakley, Problem 11936, Amer. Math. Monthly, 123 (2016), 941.

Crossrefs

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

Programs

Maple

N:= 10^9: # to get all terms <= N
R1:= {seq(2^i,i=2..ilog2(N))}:
R2:= {seq(9*2^(2*r), r=0..ilog2(floor(N/9))/2)}:
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)]))}:
sort(convert(R1 union R2 union R3, list)); # Robert Israel, Apr 25 2017

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)^kJinyuan Wang, Nov 30 2019

Extensions

a(19)-a(40) from Robert Israel, Apr 25 2017