A226353 -id:A226353 - cp's OEIS frontend

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

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

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.

			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,

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

Cf. A226353.
Cf. A226353, A226224.
Cf. digital sums for digits at various powers: A007953, A003132, A055012, A055013, A055014, A055015.

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")

cp's OEIS Frontend

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

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