A124053 Numbers n that can be expressed as the sum of the digits of both m^k and k^m for distinct numbers m and k which are not both equal to powers of 10.

7, 18, 45, 61, 72, 85, 90, 145, 270, 306, 315, 367, 376, 448, 477, 540, 547, 585, 667, 733, 756, 765, 943, 1152, 1377, 1899, 1971, 2106, 2133, 2155, 2215, 2224, 2349, 2628, 2822, 2871, 2968, 3123, 3139, 3181, 3204, 3355, 3546, 3553, 3775, 3780, 4131, 4455

Offset

1,1

Comments

If "sumdigit" denotes the sum of the digits of a number then these are the numbers n such that n=sumdigit(m^k)=sumdigit(k^m).

Two banal cases are not considered: 1) m=k because m^k=k^m and the sum of the digits is automatically equal for both the numbers; 2) powers of 10 because sumdigit(10^a)=1 for any integer a. The same number can be generated by different pairs: 477 cames from sumdigit(54^63)=sumdigit(63^54) and sumdigit(90^120)=sumdigit(120^90) 2349 cames from sumdigit(216^222)=sumdigit(222^216), sumdigit(216^225)=sumdigit(225^216) and sumdigit(219^222)=sumdigit(222^219)

Links

Table of n, a(n) for n=1..48.

Example

270 = sumdigit(36^39) = sumdigit(39^36);
1152 = sumdigit(114^126) = sumdigit(126^114);
2133 = sumdigit(204^213) = sumdigit(213^204).

Maple

P:=proc(n)local i, j, k, w, x, y; for i from 1 by 1 to n do for j from 1 by 1 to n do w:=0; x:=0; k:=i^j; y:=j^i; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; while y>0 do x:=x+y-trunc(y/10)*10; y:=trunc(y/10); od; if (w=x) and (w<>1) and (i<j) then print(i, j, w); fi; od; od; end: P(500);

Crossrefs

Cf. A124359, A124360, A046019, A124365, A124366, A124367.

Sequence in context: A133673 A023166 A002764 * A356042 A324900 A343545

Adjacent sequences: A124050 A124051 A124052 * A124054 A124055 A124056

Keyword

nonn,base

Author

Paolo P. Lava and Giorgio Balzarotti, Nov 03 2006, Nov 29 2006

Status

approved