|
| |
|
|
A337670
|
|
Numbers that can be expressed as both Sum x^y and Sum y^x where the x^y are not equal to y^x for any (x,y) pair and all (x,y) pairs are distinct.
|
|
2
|
|
|
|
432, 592, 1017, 1040, 1150, 1358, 1388, 1418, 1422, 1464, 1554, 1612, 1632, 1713, 1763, 1873, 1889, 1966, 1968, 1973, 1990, 2091, 2114, 2190, 2291, 2320, 2364, 2451, 2589, 2591, 2612, 2689, 2697, 2719, 2753, 2775, 2803, 2813, 2883, 3087, 3127, 3141, 3146
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Numbers m of form m = Sum_{i=1...k} b_i^e_i = Sum_{i=1...k} e_i^b_i such that b_i^e_i != e_i^b_i, b_i > 1, e_i > 1, k = |{{b_i, e_i}, i = 1, 2, ...}|, k > 1.
Terms of the sequence relate to the Diophantine equation Sum_{i=1...k} x_i = 0, k > 1, x_i != 0, where x_i = (b_i^e_i - e_i^b_i) such that b_i > 1, e_i > 1 and (i != j) => ({b_i, e_i} != {b_j, e_j}). That is, we are observing linear combinations of elements from {(r^n - n^r) : n,r > 1} \ {0}, under given conditions.
For sums with k = 20 terms, one infinite family of examples is known: "2^(2t) + t^(4) + 2^(2t+8) + (t+4)^(4) + 2^(2t+16) + (t+8)^(4) + 2^(2t+32) + (t+16)^(4) + 2^(2t+34) + (t+17)^(4) + 4^(t+1) + (2t+2)^(2) + 4^(t+2) + (2t+4)^(2) + 4^(t+10) + (2t+20)^(2) + 4^(t+14) + (2t+28)^(2) + 4^(t+18) + (2t+36)^(2)" is a term of the sequence, for every t > 4.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
17 = 2^3 + 3^2 = 3^2 + 2^3 is not in the sequence because {2,3} = {3,2} are not distinct.
25 = 3^3 + 2^4 = 3^3 + 4^2 is not in the sequence because 3^3 = 3^3 and 2^4 = 4^2 are commutative.
The smallest term of the sequence is:
a(1) = 432 = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7
= 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2.
The smallest term that has more than one representation is:
a(11) = 1554 = 3^2 + 7^2 + 6^3 + 2^8 + 4^5
= 2^3 + 2^7 + 3^6 + 8^2 + 5^4,
a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6
= 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3.
Smallest terms with k = 5, 6, 7, 8, 9, 10 summands are:
a(9) = 1422 = 5^2 + 7^2 + 9^2 + 3^5 + 4^5
= 2^5 + 2^7 + 2^9 + 5^3 + 5^4,
a(1) = 432 = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7
= 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2,
a(2) = 592 = 3^2 + 5^2 + 7^2 + 4^3 + 2^6 + 5^3 + 2^8
= 2^3 + 2^5 + 2^7 + 3^4 + 6^2 + 3^5 + 8^2,
a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6
= 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3,
a(14) = 1713 = 3^2 + 2^5 + 6^2 + 8^2 + 4^3 + 2^7 + 3^5 + 2^9 + 5^4
= 2^3 + 5^2 + 2^6 + 2^8 + 3^4 + 7^2 + 5^3 + 9^2 + 4^5,
a(28) = 2451 = 3^2 + 5^2 + 6^2 + 8^2 + 3^4 + 2^7 + 6^3 + 3^5 + 5^4 + 2^10
= 2^3 + 2^5 + 2^6 + 2^8 + 4^3 + 7^2 + 3^6 + 5^3 + 4^5 + 10^2.
|
|
|
CROSSREFS
|
Cf. A337671 (subsequence for k <= 5).
Cf. A005188 (perfect digital invariants).
Cf. Nonnegative numbers of the form (r^n - n^r), for n,r > 1: A045575.
Cf. Numbers of the form (r^n - n^r): A024012 (r = 2), A024026 (r = 3), A024040 (r = 4), A024054 (r = 5), A024068 (r = 6), A024082 (r = 7), A024096 (r = 8), A024110 (r = 9), A024124 (r = 10), A024138 (r = 11), A024152 (r = 12).
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
