A003321 Smallest n-th order perfect digital invariant or PDI: smallest number > 1 equal to sum of n-th powers of its digits, or 0 if no such number exists.

2, 0, 153, 1634, 4150, 548834, 1741725, 24678050, 146511208, 4679307774, 32164049650, 0, 564240140138, 28116440335967, 0, 4338281769391370, 233411150132317, 0, 1517841543307505039, 63105425988599693916

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• nice

Comments

Except for the initial term, this is the third column of A252648. - M. F. Hasler, Feb 16 2015

a(n) = 0 if n>1 and in A262094. - Dmitry Kamenetsky, Jun 05 2020

Examples

			1^3 + 5^3 + 3^3 = 153.
1*0^17 + 5*1^17 + 2*2^17 + 4*3^17 + 1*4^17 + 1*5^17 + 1*7^17 = 233411150132317.
		

References

• M. Gardner, The Magic Numbers of Dr Matrix. Prometheus, Buffalo, NY, 1985, p. 249.
• J. S. Madachy, Mathematics on Vacation, Thomas Nelson and Sons Ltd. 1966, p. 164.
• J. S. Madachy, Madachy's Mathematical Recreations, Dover, p. 164.
• C. A. Pickover, Keys to Infinity. New York: W. H. Freeman, pp. 169-170, 1995.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Joseph Myers, Table of n, a(n) for n=1..109
• L. E. Deimel, Narcissistic Numbers
• H. Heinz, Narcissistic Numbers
• Eric Weisstein's World of Mathematics, Narcissistic Number.

Crossrefs

Cf. A001694, A007532, A005934, A005188, A014576, A023052, A046074, A046761.

In other bases: A033835 (base 3), A033836 (base 4), A033837 (base 5), A033838 (base 6), A033839 (base 7), A033840 (base 8), A033841 (base 9).

Programs

PARI

a(n)=m=1;while(m*9^n>=10^m,m++);for(k=2,10^m,d=digits(k);s=sum(i=1,#d,d[i]^n);if(s==k,return(k)));0
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Dec 19 2014

Extensions

Additional comments from Lekraj Beedassy, May 23 2001

Extended and cross-references edited by Joseph Myers, Jun 28 2009