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A255668 Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits. 3
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 (list; graph; refs; listen; history; text; internal format)
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 A318486 A303850

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

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Last modified September 23 17:19 EDT 2021. Contains 347618 sequences. (Running on oeis4.)