OFFSET
1,3
COMMENTS
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.
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.
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.
For n = 21, 23, and 24, a(n) > 10^10.
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.
For n in {51, 52, 53, ..., 64, 65} and {67, 68, 69, ..., 73, 74}, a(n) > 10^11.
For n in {75, 76, 77} and {79, 80, 81, ..., 96, 97, 99}, a(n) > 5*10^11.
A few nonzero terms were added by math4pad.net @PascalCardin
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
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..500 (n = 1..100 from Hiroaki Yamanouchi)
John D. Cook, Monday morning math puzzle (2012).
Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
Dean Morrow, Cycles of a family of digit functions
EXAMPLE
12 is the smallest number such that 3^1 + 3^2 = 12 so a(3) = 12.
4624 is the smallest number such that 4^4 + 4^6 + 4^2 + 4^4 = 4624 so a(4) = 4624.
1033 is the smallest number such that 8^1 + 8^0 + 8^3 + 8^3 = 1033 so a(8) = 1033.
PROG
(PARI)
Min(n)=for(k=0, oo, if(n+k<=10^k, return(10^k)))
Max(n)=for(k=1, oo, if(k*n^9<=10^k-1, return(10^(k-1))))
a(n)={for(k=Min(n), Max(n), my(d=digits(k)); if(sum(i=1, #d, n^d[i])==k, return(k))); 0}
{ for(n=1, 100, print1(a(n), ", ")) } \\ Derek Orr, Aug 01 2014; corrected by Jason Yuen, Feb 25 2025
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Derek Orr, Jan 19 2014
EXTENSIONS
More terms and edited extensively by Derek Orr, Aug 26 2014
a(21)-a(30) from Hiroaki Yamanouchi, Sep 27 2014
STATUS
approved