A244212 Numbers n for which the alternating sum of the digits of n^n is 0.

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

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.

n such that A000312(n) is in A135499. - Robert Israel, Jul 13 2014

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

Jens Kruse Andersen and Robert G. Wilson v, Table of n, a(n) for n = 1..139 (a(48) to a(63) from Jens Kruse Andersen).

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;
select(filter, 11 * [$1..1000]); # Robert Israel, Jul 13 2014

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

Cf. A000312, A065816, A135499, A244144.

Sequence in context: A324486 A351170 A122502 * A303582 A289015 A063302

Adjacent sequences: A244209 A244210 A244211 * A244213 A244214 A244215

Keyword

nonn,base

Author

Anthony Sand, Jun 23 2014

Extensions

a(9)-a(24) from Michel Marcus, Jun 23 2014

a(25)-a(44) from Robert G. Wilson v, Jul 13 2014

Status

approved