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A285438
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Perfect powers that are also the sum of two powers of a prime p.
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1
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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
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internal format)
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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324 = 18^2 = 3^4 + 3^5.
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MAPLE
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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
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PROG
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(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); }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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