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A236067 a(n) is the least number m such that m = n^d_1 + n^d_2 + ... + n^d_k where d_k represents the k-th digit in the decimal expansion of m, or 0 if no such number exists.

%I A236067 #109 Mar 20 2025 10:01:15
%S A236067 1,0,12,4624,3909511,0,13177388,1033,10,0,0,0,0,0,2758053616,1053202,
%T A236067 7413245658,419370838921,52135640,1347536041,833904227332,5117557126,
%U A236067 3606012949057,5398293152472,31301,0,15554976231978,405287637330,35751665247,19705624111111
%N A236067 a(n) is the least number m such that m = n^d_1 + n^d_2 + ... + n^d_k where d_k represents the k-th digit in the decimal expansion of m, or 0 if no such number exists.
%C A236067 The 0's in the sequence are definite. There exists both a maximum and a minimum number that a(n) can be based on n. They are given in the programs below as Max(n) and Min(n), respectively.
%C A236067 It is known that a(22) = 5117557126, a(25) = 31301, a(29) = 35751665247, a(32) = 2112, a(33) = 1224103, a(37) = 111, a(40) = 102531321, a(48) = 25236435456, a(50) = 101, a(66) = 2524232305, a(78) = 453362316342, a(98) = 100, and a(100) = 20102.
%C A236067 There are an infinite number of nonzero entries. First, note if a(n) is nonzero, a(n) >= n. Further, a(9) = 10, a(98) = 100, a(997) = 1000, ..., a(10^k-k) = 10^k for all k >= 0.
%C A236067 For n = 21, 23, and 24, a(n) > 10^10.
%C A236067 For n in {26, 27, 28, 30, 31, 34, 35, 36, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49}, a(n) > 5*10^10.
%C A236067 For n in {51, 52, 53, ..., 64, 65} and {67, 68, 69, ..., 73, 74}, a(n) > 10^11.
%C A236067 For n in {75, 76, 77} and {79, 80, 81, ..., 96, 97, 99}, a(n) > 5*10^11.
%C A236067 A few nonzero terms were added by math4pad.net @PascalCardin
%C A236067 a(1000) = 1000000000000002002017, a(10000) = 0, a(1000000) = 1000002000010, a(10000000) = 200000020000011. It looks like a(10^k) in decimal consists of mostly the digits 0, 1 and 2. - _Chai Wah Wu_, Dec 07 2017
%H A236067 Chai Wah Wu, <a href="/A236067/b236067.txt">Table of n, a(n) for n = 1..500</a> (n = 1..100 from Hiroaki Yamanouchi)
%H A236067 John D. Cook, <a href="http://www.johndcook.com/blog/2012/02/20/monday-morning-math-puzzle/">Monday morning math puzzle</a> (2012).
%H A236067 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_21">Digital Invariants and Narcissistic Numbers</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
%H A236067 Dean Morrow, <a href="https://citeseerx.ist.psu.edu/pdf/334939488b316538371244a43a3538d403faa190">Cycles of a family of digit functions</a>
%e A236067 12 is the smallest number such that 3^1 + 3^2 = 12 so a(3) = 12.
%e A236067 4624 is the smallest number such that 4^4 + 4^6 + 4^2 + 4^4 = 4624 so a(4) = 4624.
%e A236067 1033 is the smallest number such that 8^1 + 8^0 + 8^3 + 8^3 = 1033 so a(8) = 1033.
%o A236067 (PARI)
%o A236067 Min(n)=for(k=0,oo,if(n+k<=10^k,return(10^k)))
%o A236067 Max(n)=for(k=1,oo,if(k*n^9<=10^k-1,return(10^(k-1))))
%o A236067 a(n)={for(k=Min(n), Max(n), my(d=digits(k)); if(sum(i=1,#d,n^d[i])==k, return(k))); 0}
%o A236067 { for(n=1, 100,print1(a(n), ", ")) } \\ _Derek Orr_, Aug 01 2014; corrected by _Jason Yuen_, Feb 25 2025
%Y A236067 Cf. A139410 (for 4th term), A003321, A296138, A296139.
%K A236067 nonn,base
%O A236067 1,3
%A A236067 _Derek Orr_, Jan 19 2014
%E A236067 More terms and edited extensively by _Derek Orr_, Aug 26 2014
%E A236067 a(21)-a(30) from _Hiroaki Yamanouchi_, Sep 27 2014