OFFSET
1,1
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
LINKS
Robert Israel, Table of n, a(n) for n = 1..6655
W. Weakley, Problem 11936, Amer. Math. Monthly, 123 (2016), 941.
EXAMPLE
324 = 18^2 = 3^4 + 3^5.
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
PROG
(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); }
upto(10^9) \\ Jinyuan Wang, Nov 30 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Josephy, Apr 18 2017
EXTENSIONS
a(19)-a(40) from Robert Israel, Apr 25 2017
STATUS
approved