A298591 - OEIS

%I #22 Sep 18 2018 09:06:38

%S 12,13,17,20,24,25,29,31,33,34,35,36,40,41,43,44,45,48,52,53,57,58,59,

%T 61,63,65,68,72,73,74,76,80,81,85,89,90,91,96,97,100,104,106,108,109,

%U 113,116,117,125,127,129,130,132,133,134,136,137,141,144,145,146,148,149,150

%N Numbers which are the sum of two distinct perfect powers x^k + y^m with x, y, k, m >= 2.

%C The number of terms between 2^(n-1) and 2^n-1 is, for n = 1, 2, 3, ...: 0, 0, 0, 2, 6, 17, 24, 69, 129*, 215, 425, 891, 1571, 2994, 5655*, 10535, 20132, 38840, 73510, 140730, 268438*, 514262, ... (For terms with * the next larger power of 2 is in the sequence, so it would be, e.g., ..., 130, 214, ... if we count from 2^n+1 to 2^(n+1).) At 2^22 this corresponds to a density of about 25%, decreasing by about 1% at each power of 2.

%H Robert Israel, <a href="/A298591/b298591.txt">Table of n, a(n) for n = 1..10000</a>

%H Andrew Lohr, <a href="https://arxiv.org/abs/1805.00076">Several Topics in Experimental Mathematics</a>, arXiv:1805.00076 [math.CO], 2018.

%e 12 = 2^2 + 2^3, 13 = 2^2 + 3^2, 17 = 2^3 + 3^2, ...

%p N:= 1000: # for all terms <= N

%p PP:= {seq(seq(x^k,k=2..floor(log[x](N))),x=2..floor(sqrt(N)))}:

%p sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],i=1..j-1),j=2..nops(PP))},N),list)); # _Robert Israel_, May 27 2018

%t max = 150; Table[If[x^k == y^m, Nothing, x^k + y^m], {x, 2, Sqrt[max-4]}, {y, x, Sqrt[max-4]}, {k, 2, Log[2, max-4]}, {m, 2, Log[2, max-4]}] // Flatten // Select[#, # <= max &]& // Union (* _Jean-François Alcover_, Sep 18 2018 *)

%o (PARI) is(n,A=A076467,s=sum2sqr(n))={for(i=1,#s, vecmin(s[i])>1 && s[i][1]!=s[i][2] && return(1)); for(i=2,#A, n>A[i]||return; ispower(n-A[i]) && A[i]*2!=n && return(1)) \\ A076467 must be computed up to limit n.

%Y Cf. A076467, A111231, A304433, A304434, A304435, A304436.

%K nonn

%O 1,1

%A _M. F. Hasler_, May 26 2018