A255668 Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits.

1, 10, 2, 6, 5, 8, 3, 7, 5, 6, 3, 10, 2, 3, 3, 2, 4, 6, 2, 6, 3, 4, 2, 7, 5, 10, 2, 9, 2, 9, 2, 6, 3, 5, 3, 6, 3, 5, 5, 7, 2, 2, 4, 9, 6, 9, 5, 7, 2, 3, 2, 4, 2, 3, 6, 4, 5, 4, 2, 4, 4, 4, 3, 7, 3, 6, 3, 4, 3, 3, 4, 3, 4, 5, 3, 4, 5, 5, 3, 3, 2, 3, 2, 4, 3, 8, 3, 5, 2, 7, 3

Offset

0,2

Comments

Row lengths of the table A252648.

For a number with d digits, the sum of n-th powers cannot exceed d*9^n, but the number is not less than 10^(d-1). Therefore there is only a finite number of possible perfect digital invariants for any n, the largest of which has at most d* digits, where d* = 1+(n*log(9)+log d*)/log(10).

Links

Don Knuth, Table of n, a(n) for n = 0..172

Table of a(n) for n=0..172 [From Don Knuth, Sep 09 2015]

Formula

a(n) >= 2 for all n > 0, since 0 and 1 are digital invariants for any power n > 0.

Example

a(0)=1 because 1 is the only number equal to the sum of 0th powers of its digits.
a(1)=10 because { 0, 1, ... 9 } are the only numbers equal to the sum of their digits (taken to the power 1).
a(2)=2 because 0 and 1 are the only numbers equal to the sum of the squares of their digits.
a(3)=6 because { 0, 1, 153, 370, 371, 407 } is the set of all numbers equal to the sum of the 3rd powers of their digits, cf. A046197.
For more examples, see the table A252648.

Mathematica

Reap@ For[n = 0, n < 6, n++, Sow@ Length@ Select[Range[0, 10^(n + 1)], Plus @@ (IntegerDigits[#]^n) == # &]] // Flatten // Rest (* Michael De Vlieger, Apr 14 2015 *)

Crossrefs

Cf. A252648, A003321, A046197, A052455, A052464, A124068, A124069, A226970.

Sequence in context: A299980 A037922 A111287 * A187815 A366197 A318486

Adjacent sequences: A255665 A255666 A255667 * A255669 A255670 A255671

Keyword

nonn,base

Author

M. F. Hasler, Apr 14 2015

Extensions

a(10)-a(90) from Don Knuth, Sep 09 2015

Status

approved