A298591 Numbers which are the sum of two distinct perfect powers x^k + y^m with x, y, k, m >= 2.

12, 13, 17, 20, 24, 25, 29, 31, 33, 34, 35, 36, 40, 41, 43, 44, 45, 48, 52, 53, 57, 58, 59, 61, 63, 65, 68, 72, 73, 74, 76, 80, 81, 85, 89, 90, 91, 96, 97, 100, 104, 106, 108, 109, 113, 116, 117, 125, 127, 129, 130, 132, 133, 134, 136, 137, 141, 144, 145, 146, 148, 149, 150

Offset: 1

Author

M. F. Hasler, May 26 2018

Keywords

• nonn

Comments

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.

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Andrew Lohr, Several Topics in Experimental Mathematics, arXiv:1805.00076 [math.CO], 2018.

Crossrefs

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

Programs

Maple

N:= 1000: # for all terms <= N
PP:= {seq(seq(x^k,k=2..floor(log[x](N))),x=2..floor(sqrt(N)))}:
sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],i=1..j-1),j=2..nops(PP))},N),list)); # Robert Israel, May 27 2018

Mathematica

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 *)

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. See A133388 for sum2sqr.