|
|
A244212
|
|
Numbers n for which the alternating sum of the digits of n^n is 0.
|
|
2
|
|
|
22, 55, 77, 99, 132, 187, 286, 1056, 1463, 1474, 1606, 1837, 2277, 2981, 4785, 4851, 5313, 5588, 5929, 7227, 8272, 8415, 8492, 8954, 11517, 12573, 12628, 13156, 14883, 15972, 17688, 22066, 23936, 24915, 25850, 27522, 34045, 36289, 36806, 38489, 40744, 43450, 46794, 48092
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The result of alternately adding and subtracting the digits of n sometimes differs in sign when the procedure goes from left to right or right to left. For example, if n = 1234, 1 - 2 + 3 - 4 = -2, whereas 4 - 3 + 2 - 1 = +2. However, if the sum is zero when adding and subtracting from left to right, it will also be zero when adding and subtracting from right to left.
All terms are multiples of 11. This follows from the divisibility rule for 11. - Jens Kruse Andersen, Jul 13 2014
Number of terms less than 10^k: 0, 4, 7, 24, 55, 135, ..., . - Robert G. Wilson v, Jul 18 2014
Numbers for which the alternating sum of the digits of n^n are == 0 (Mod 10): 12, 22, 23, 35, 45, 46, 47, 55, 57, 77, 99, 117, 126, 132, 151, ..., . Obviously the members of A244212 are included here. - Robert G. Wilson v, Jul 20 2014
|
|
LINKS
|
|
|
FORMULA
|
s = 0; m = 1; for digit[n,i=1..j] of n, s = s + digit[i] * m; m = -m; next i; if s = 0, print n;
|
|
EXAMPLE
|
22^22 = 341427877364219557396646723584, therefore the alternating sum = 4 - 8 + 5 - 3 + 2 - 7 + 6 - 4 + 6 - 6 + 9 - 3 + 7 - 5 + 5 - 9 + 1 - 2 + 4 - 6 + 3 - 7 + 7 - 8 + 7 - 2 + 4 - 1 + 4 - 3 = 0.
|
|
MAPLE
|
filter:= proc(n) local x, j;
x:= convert(n^n, base, 10);
evalb(add((-1)^j*x[j], j=1..nops(x)) = 0)
end proc;
|
|
MATHEMATICA
|
fQ[n_] := Block[{id = IntegerDigits[ n^n]}, Sum[ id[[i]]*(-1)^i, {i, Length@ id}] == 0]; k = 11; lst = {}; While[k < 100001, If[ fQ@ k, AppendTo[ lst, k]; Print@ k]; k+= 11]; lst (* Robert G. Wilson v, Jul 13 2014 *)
|
|
PROG
|
(PARI) isok(n) = d = digits(n^n) ; sum(i=1, #d, d[i]*(-1)^i) == 0; \\ Michel Marcus, Jun 25 2014
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,base
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|