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%I #31 Dec 26 2021 01:30:34
%S 1,2,3,4,5,6,7,8,9,20,22,102,120,201,210
%N Numbers k with the property that (sum of digits of k) times (number of digits of k) equals (sum of {each digit of k} raised to the power {number of digits in k}).
%C Numbers with the property that q-p = 0, where p = (sum of digits of k) times (number of digits of k) (A110805) and q = (sum of {each digit of k} raised to the power {number of digits in k}) (A101337).
%C From _David Consiglio, Jr._, Jan 29 2017: (Start)
%C The sequence is finite and 210 is the last term. Proof:
%C 1. Let Y = length(k).
%C 2. The maximum value of digit_sum(k) is 9Y.
%C 3. Since p = Y*digit_sum(k), p has a maximum value of 9Y^2.
%C 4. 2^Y > 9Y^2 for all Y > 9. Therefore q > p for all numbers longer than 9 digits that contain any digits > 1.
%C a. Example: 1,000,000,002. q = 1^5 + 8*0^5 + 2^10 = 1025. The largest p value for a 10-digit number would be for 9,999,999,999 which has p = 10*(9*10) = 900. Since q > all possible p values at this size, any term of this sequence must either have fewer than 10 digits or contain only 0's and 1's as digits.
%C 5. Now we can consider only numbers with 0 and 1 digits.
%C 6. Let Z = number of 1 digits in a number k.
%C 7. q = Z because q = Z*1^Y + (Y-Z)*0^Y = Z.
%C 8. p = YZ because the sum of the digits is equal to the number of 1's.
%C 9. Z = YZ only in the case of Y = 1. Thus, the only term of this sequence that contains only 0's and 1's has a length of only 1 digit. Thus, k = 1 is in this sequence.
%C 10. Therefore a candidate number must have fewer than 10 digits if it contains a digit 2 or larger, and must have fewer than 2 digits if it does not. All numbers in this range have been checked, and no additional values of k with q = p have been found.
%C Thus the sequence is finite. (End)
%e 9 is a term because 9^1 = 1*(9);
%e 20 is a term because 2^2 + 0^2 = 2*(2 + 0);
%e 210 is a term because 2^3 + 1^3 + 0^3 = 3*(2 + 1 + 0).
%t W = Array[Total[IntegerDigits[#]^IntegerLength[#]]&, 100]-Table[IntegerLength[n] * Total[IntegerDigits[n]], {n, 100}]; Flatten[Position[W,0]]
%Y Cf. A101337, A110805.
%K nonn,base,fini
%O 1,2
%A _José de Jesús Camacho Medina_, Jan 28 2017
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