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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 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=0 & av=0 & bv

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