A226352 Number of integers k in base n whose squared digits sum to sqrt(k).

1, 3, 2, 2, 1, 1, 4, 2, 1, 2, 3, 6, 1, 6, 3, 3, 1, 2, 2, 3, 2, 4, 4, 4, 2, 9, 2, 4, 2, 3, 1, 3, 3, 3, 3, 1, 2, 4, 5, 4, 1, 6, 1, 5, 2, 5, 2, 5, 4, 1, 3, 5, 1, 5, 2, 5, 1, 7, 3, 2, 2, 7, 3, 2, 2, 4, 3, 2, 1, 3, 3, 6, 3, 3, 2, 1, 2, 5, 3, 4, 1, 4, 1, 3, 2, 3, 1

Offset

2,2

Comments

Any d-digit number in base n meeting the criterion must also meet the condition d*(n-1)^2 < n^(d/2). Numerically, it can be shown this limits the candidate values to squares < 22*n^4. The larger values are statistically unlikely, and in fact the largest value of k in the first 1000 bases is ~9.96*n^4 in base 775.

Links

Christian N. K. Anderson, Table of n, a(n) for n = 2..1000

Example

In base 8, the four solutions are the values {1,16,256,2601}, which are written as {1,20,400,5051} in base 8 and
sqrt(1)    =  1 = 1^2;
sqrt(16)   =  4 = 2^2 + 0^2;
sqrt(256)  = 16 = 4^2 + 0^2 + 0^2;
sqrt(2601) = 51 = 5^2 + 0^2 + 5^2 + 1^2,

Prog

(R) inbase=function(n, b) { x=c(); while(n>=b) { x=c(n%%b, x); n=floor(n/b) }; c(n, x) }
for(n in 2:50) cat("Base", n, ":", which(sapply((1:(4.7*n^2))^2, function(x) sum(inbase(x, n)^2)==sqrt(x)))^2, "\n")

Crossrefs

Cf. A226353.

Cf. A226353, A226224.

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

Sequence in context: A170822 A054546 A065310 * A238402 A016558 A154395

Adjacent sequences: A226349 A226350 A226351 * A226353 A226354 A226355

Keyword

nonn,base

Author

Christian N. K. Anderson and Kevin L. Schwartz, Jun 04 2013

Status

approved