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.

1, 0, 12, 4624, 3909511, 0, 13177388, 1033, 10, 0, 0, 0, 0, 0, 2758053616, 1053202, 7413245658, 419370838921, 52135640, 1347536041, 833904227332, 5117557126, 3606012949057, 5398293152472, 31301, 0, 15554976231978, 405287637330, 35751665247, 19705624111111

Offset: 1

Author

Derek Orr, Jan 19 2014

Keywords

• nonn
• base

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

Examples

			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.
		

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

Crossrefs

Cf. A139410 (for 4th term), A003321, A296138, A296139.

Programs

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

Extensions

More terms and edited extensively by Derek Orr, Aug 26 2014

a(21)-a(30) from Hiroaki Yamanouchi, Sep 27 2014