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A209242
The largest fixed value (neither happy nor sad) in base n.
3
8, 1, 18, 1, 45, 52, 50, 1, 72, 125, 160, 1, 128, 1, 261, 260, 200, 1, 425, 405, 490, 1, 338, 1, 657, 628, 450, 848, 936, 845, 1000, 832, 648, 1, 1233, 1377, 800, 1, 1450, 1445, 1813, 1341, 1058, 1856, 2125, 1844, 1250, 1525, 1352, 2205, 2560, 1, 2873, 1, 3200
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
Cf. A193583.
Sequence in context: A013615 A359628 A369404 * A103884 A103883 A317640
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
Program improved and sequence extended by Christian N. K. Anderson, Apr 25 2013.
STATUS
approved