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User: Michael Josephy

Michael Josephy has authored 1 sequences.

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

Michael Josephy, Apr 18 2017

nonn

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

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

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

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

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

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

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

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User: Michael Josephy

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

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