OFFSET
3,1
COMMENTS
A number is a fixed value if it is the sum of its own squared digits. Such values >1 are the only numbers that are neither happy (A007770) nor unhappy (A031177) in that base.
The number of fixed values in base B (A193583) is equal to one less than the number of divisors of (1+B^2) (Beardon, 1998, Theorem 3.1).
No fixed point has more than 2 digits in base B, and the two-digit number a+bB must satisfy the condition that (2a-1)^2+(2b-B)^2=1+B^2 (Beardon, 1998, Theorem 2.5). Since there are a finite number of ways to express 1+B^2 as the sum of two squares (A002654), this limits the search space.
Because fixed points have a maximum value of B^2-1 in base B, there are a large number of solutions near perfect squares, x^2. Surprisingly, there are also a large number of points near "half-squares", (x+.5)^2. See "Ulam spiral" in the links.
LINKS
Christian N. K. Anderson, All fixed values in base n for n=3..10000
Christian N. K. Anderson, Ulam spiral of maximum fixed values in base n for=3..1000
Alan F. Beardon, Sums of Squares of Digits, The Mathematical Gazette, 82(1998), 379-388.
EXAMPLE
a(7)=45 because in base 7, 45 is 63 and 6^2+3^2=45. The other fixed values in base 7 are 32, 25, 10 and (as always) 1.
PROG
(R) #ya=number of fixed points, yb=values of those fixed points
library(gmp); ya=rep(0, 200); yb=vector("list", 200)
for(B in 3:200) {
w=1+as.bigz(B)^2
ya[B]=prod(table(as.numeric(factorize(w)))+1)-1
x=1; y=0; fixpt=c()
if(ya[B]>1) {
while(2*x^2<w) {
if(issquare((y=as.numeric(w-x^2)))) {
y=sqrt(y)
av=(1+rep(c(-1, -1, 1, 1), 2)*rep(c(x, y), e=4))/2
bv=(B+rep(c(-1, 1), 4)*rep(c(y, x), e=4))/2
ix=av>=0 & av<B & bv>=0 & bv<B & !(av==0 & bv==0) & isint(av)
fixpt=c(fixpt, (av+B*bv)[ix])
}
x=x+1
}
} else fixpt=1
yb[[B]]=sort(unique(fixpt))
}
sapply(yb, max)
(Python)
from sympy.ntheory.digits import digits
def ssd(n, b): return sum(d**2 for d in digits(n, b)[1:])
def a(n):
m = n**2 - 1
while m != ssd(m, n): m -= 1
return m
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Aug 01 2021
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Christian N. K. Anderson, Apr 22 2013
EXTENSIONS
Program improved and sequence extended by Christian N. K. Anderson, Apr 25 2013.
STATUS
approved