proposed

approved

editing

proposed

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 < 22n22*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.

Christian N. K. Anderson, <a href="/A226353/a226353.txt">Table of base, all solutions in base 10, and all solutions in base n, </a> for bases 2 to 1000.</a>

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

(R) inbase=function(n, b) { x=c(); while(n>=b) { x=c(n%%b, x); n=floor(n/b) }; c(n, x) }

a(n)=1 iff A226352(n)=1.

In base 8, the four solutions are the values {1,16,256,2601}, which are written as {1,20,400,5051} in base 8 andsqrt(1)=1=1^2and

sqrt(1)=1=1^2

Cf. A226352, A226224.

Cf. A226353, A226224.

Cf. digital root of n: A010888.

Cf. A226353, A226224.

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

Cf. A226224.

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

Cf. digital root of n: A010888.

Christian N. K. Anderson, <a href="/A226353/a226353.txt">Table of base, all solutions in base 10, and all solutions in base n, for bases 2 to 1000.</a>

Christian N. K. Anderson, <a href="/A226353/b226353.txt">Table of n, a(n) for n = 2..1000</a>

allocated for Kevin L. Schwartz

Largest integer k in base n whose squared digits sum to sqrt(k).

1, 49, 169, 36, 1, 1, 2601, 1089, 1, 8836, 33489, 44100, 1, 149769, 128164, 96721, 1, 156816, 1225, 40804, 12321, 831744, 839056, 1149184, 1737124, 3655744, 407044, 1890625, 2208196, 1089, 1, 1466521, 6125625, 2235025, 2832489, 1, 3759721, 6885376, 8844676

2,2

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 < 22n^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 andsqrt(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

(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")

Cf. A226352.

allocated

nonn,base

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

approved

editing

allocated for Kevin L. Schwartz

allocated

approved