A226087 Number of values k in base n for which the sum of digits of k = sqrt(k).

1, 4, 2, 3, 3, 6, 2, 2, 2, 5, 2, 6, 2, 5, 5, 2, 2, 4, 2, 6, 6, 4, 2, 5, 2, 4, 2, 6, 2, 11, 2, 2, 6, 4, 5, 6, 2, 4, 6, 5, 2, 11, 2, 6, 5, 4, 2, 6, 2, 4, 6, 5, 2, 4, 5, 5, 6, 4, 2, 13, 2, 4, 4, 2, 5, 11, 2, 5, 6, 11, 2, 5, 2, 4, 6, 6, 6, 11, 2, 5, 2, 4, 2, 12, 5

Offset: 2

Author

Christian N. K. Anderson and Kevin L. Schwartz, May 25 2013

Keywords

• nonn
• base

Comments

Values of k in base n have at most 3 digits. Proof: Because sqrt(k) increases faster than the digit sum of k, only numbers with d digits meeting the condition d*(n-1) >= n^(d/2) are candidate fixed points. d < 3 for n > 6, and since there are no fixed points of four or more digits in bases 2 through 5, there are no fixed points in any base with more than 3 digits.

From the above, it can be shown that for three-digit fixed points of the form xyz, x <= 6; also x <= 4 for n > 846. These theoretical upper limits are statistically unlikely, and in fact of the 86356 solutions in bases 2 to 10000, only 6.5% of them begin with 2, and none begin with 3 through 6.

Examples

			For a(16)=5 the solutions are the square numbers {1, 36, 100, 225, 441} because in base 16 they are written as {1, 24, 64, E1, 1B9} and
  sqrt(1) = 1
  sqrt(36) = 6 = 2+4
  sqrt(100) = 10 = 6+4
  sqrt(225) = 15 = 14+1, and
  sqrt(441) = 21 = 1+11+9.
		

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A226224.

Cf. digital sums for digits at various powers: A007953, A003132, A055012, A055013, A055014, A055015.

Programs

R

sapply(2:16,function(n) sum(sapply((1:(n^ifelse(n>6,1.5,2)))^2, function(x) sum(inbase(x,n))==sqrt(x))))