A046881
Smallest number that is sum of 2 positive distinct n-th powers in 2 different ways.
Original entry on oeis.org
5, 65, 1729, 635318657
Offset: 1
5 = 1^1 + 4^1 = 2^1 + 3^1;
65 = 1^2 + 8^2 = 4^2 + 7^2;
1729 = 1^3 + 12^3 = 9^3 + 10^3; etc.
- R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.
-
(* This naive program is not convenient for n > 3 *) r[n_, k_] := Reduce[0 < x < y && x^n + y^n == k, {x, y}, Integers]; a[n_] := Catch[ For[ k = 1, True, k++, rk = r[n, k]; If[rk =!= False, If[ Head[rk] == Or && Length[rk] == 2, Print["n = ", n, ", k = ", k]; Throw[k]]]]]; Table[a[n], {n, 1, 3}] (* Jean-François Alcover, Jul 30 2013 *)
A230561
Smallest number that is the sum of two positive n-th powers in >= n ways.
Original entry on oeis.org
2, 50, 87539319
Offset: 1
2 = 1^1 + 1^1.
50 = 1^2 + 7^2 = 5^2 + 5^2.
87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.
- R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.
A230562
Smallest number that is the sum of 2 positive 4th powers in >= n ways.
Original entry on oeis.org
0, 2, 635318657
Offset: 0
0 = (empty sum).
2 = 1^4 + 1^4.
635318657 = 59^4 + 158^4 = 133^4 + 134^4.
- R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, 2008; section 21.11.
A338799
Smallest number that is the sum of two n-th powers of primes in two different ways.
Original entry on oeis.org
10, 338, 6058655748, 3262811042
Offset: 1
10 = 3 + 7 = 5 + 5.
338 = 7^2 + 17^2 = 13^2 + 13^2.
6058655748 = 61^3 + 1823^3 = 1049^3 + 1699^3.
3262811042 = 7^4 + 239^4 = 157^4 + 227^4.
A338800
Smallest number that is the sum of two distinct n-th powers of primes in two different ways.
Original entry on oeis.org
16, 410, 6058655748, 3262811042
Offset: 1
16 = 3 + 13 = 5 + 11.
410 = 7^2 + 19^2 = 11^2 + 17^2.
6058655748 = 61^3 + 1823^3 = 1049^3 + 1699^3.
3262811042 = 7^4 + 239^4 = 157^4 + 227^4.
-
f:= proc(n) local S,P,p,pn,b;
S:= {}:
P:= {}:
p:= 1:
b:= infinity;
do
p:= nextprime(p);
pn:= p^n;
if pn > b then return b fi;
V:= select(`<`,map(`+`,P,pn),b);
newv:= V intersect S;
S:= S union V;
P:= P union {p^n};
if newv <> {} then
b:= min(newv);
S:= select(`<`,S,b);
P:= select(`<`,P, b);
fi;
od:
end proc:
map(f, [$1..4]); # Robert Israel, Nov 13 2020
A363759
Smallest number that can be written as a sum of a positive n-th power and a positive (n+1)-th power in 2 different ways.
Original entry on oeis.org
5, 17, 4097, 1048577, 1073741825, 4398046511105, 72057594037927937, 4722366482869645213697, 1237940039285380274899124225, 1298074214633706907132624082305025, 5444517870735015415413993718908291383297
Offset: 1
5 = 1^1 + 2^2 = 4^1 + 1^2.
17 = 3^2 + 2^3 = 4^2 + 1^3.
4097 = 1^3 + 8^4 = 16^3 + 1^4.
1048577 = 1^4 + 16^6 = 32^5 + 1^6.
-
a(n) = my(k=1); while (!(sum(j=1, k^(1/n), (k>j^n)*ispower(k-j^n, n+1))==2), k++); k;
Showing 1-6 of 6 results.
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