A209242 The largest fixed value (neither happy nor sad) in base n.

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

Author

Christian N. K. Anderson, Apr 22 2013

Keywords

• nonn
• base

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.

Examples

			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.
		

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.

Crossrefs

Cf. A007770, A031177.

Cf. A193583.

Programs

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

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
				

Extensions

Program improved and sequence extended by Christian N. K. Anderson, Apr 25 2013.