A226063 Number of fixed points in base n for the sum of the fourth power of its digits.

1, 1, 3, 4, 1, 1, 7, 3, 4, 3, 1, 2, 1, 7, 2, 2, 1, 4, 2, 6, 2, 3, 1, 3, 1, 11, 3, 3, 2, 2, 7, 4, 1, 4, 3, 1, 3, 4, 1, 2, 2, 2, 3, 4, 2, 2, 1, 2, 1, 2, 1, 2, 4, 3, 3, 2, 2, 1, 3, 2, 5, 2, 9, 2, 1, 2, 1, 1, 3, 2, 2, 1, 2, 5, 1, 5, 5, 4, 2, 5, 3, 2, 2, 3, 3, 1, 2

Offset

2,3

Comments

All fixed points in base n have at most 5 digits. Proof: In order to be a fixed point, a number with d digits in base n must meet the condition n^d <= d*(n-1)^4, which is only possible for d < 5.

For 5-digit numbers vwxyz in base n, only numbers where v*n^4 + n^3 - 1 <= v^4 + 3*(n-1)^4 or v*n^4 + n^4 - 1 <= v^4 + 4*(n-1)^4 are possible fixed points. v <= 2 for n <= 250.

Links

Christian N. K. Anderson, Table of n, a(n) for n = 2..250

Example

For a(8)=7, the solutions are {1,16,17,256,257,272,273}. In base 8, these are written as {1, 20, 21, 400, 401, 420, 421}. Because 1^4 = 1, 2^4 + 0^4 = 16, 2^4 + 1^4 = 17, 4^4 + 0^4 + 0^4 = 256, etc., these are the fixed points in base 8.

Prog

(R) inbase=function(n, b) { x=c(); while(n>=b) { x=c(n%%b, x); n=floor(n/b) }; c(n, x) }
yn=rep(NA, 20)
for(b in 2:20) yn[b]=sum(sapply(1:(1.5*b^4), function(x) sum(inbase(x, b)^4))==1:(1.5*b^4)); yn

Crossrefs

Cf. A226064 (greatest fixed point).

Cf. A052455 (fixed points in base 10).

Cf. A023052, A046074, A046197.

Sequence in context: A105283 A347178 A123019 * A326748 A204999 A294731

Adjacent sequences: A226060 A226061 A226062 * A226064 A226065 A226066

Keyword

nonn,base

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 24 2013

Status

approved