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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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=3..57.

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)

CROSSREFS

Cf. A007770, A031177.

Cf. A193583.

Sequence in context: A126000 A326992 A013615 * A103884 A103883 A317640

Adjacent sequences:  A209239 A209240 A209241 * A209243 A209244 A209245

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

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Last modified October 23 14:35 EDT 2019. Contains 328345 sequences. (Running on oeis4.)