A252648 Irregular table of perfect digital invariants for n > 1, i.e., numbers equal to the sum of n-th powers of their digits, read by rows.

1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 0, 1, 153, 370, 371, 407, 0, 1, 1634, 8208, 9474, 0, 1, 4150, 4151, 54748, 92727, 93084, 194979, 0, 1, 548834, 0, 1, 1741725, 4210818, 9800817, 9926315, 14459929, 0, 1, 24678050, 24678051, 88593477, 0, 1, 146511208, 472335975, 534494836, 912985153, 0, 1, 4679307774

Offset: 0

Author

Derek Orr, Dec 19 2014

Keywords

• nonn
• base
• tabf

Comments

The third column is listed in A003321. - M. F. Hasler, Feb 16 2015

For a number x >= 10^(d-1) with d digits, the sum of n-th powers of these digits cannot exceed d*9^n. Therefore there is only a finite number of possible perfect digital invariants for any n, the largest of which has at most d* digits, where d* = 1+(n*log(9)+log d*)/log(10). - M. F. Hasler, Apr 14 2015

Examples

			The table starts:
1; (n = 0; 1 = 1^0.)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9; (n = 1)
0, 1; (n = 2)
0, 1, 153, 370, 371, 407; (n = 3, A046197)
0, 1, 1634, 8208, 9474; (n = 4, A052455)
0, 1, 4150, 4151, 54748, 92727, 93084, 194979; (n = 5, A052464)
0, 1, 548834; (n = 6)
0, 1, 1741725, 4210818, 9800817, 9926315, 14459929; (n = 7, A124068)
0, 1, 24678050, 24678051, 88593477; (n = 8, A124069)
0, 1, 146511208, 472335975, 534494836, 912985153; (n = 9, A226970)
The third column corresponds to A003321.
The term 153 is member of the row n=3 because 153 = 1^3 + 5^3 + 3^3. - _M. F. Hasler_, Apr 14 2015
		

Links

• Don Knuth, Table of a(n) for n=0..732
• Don Knuth, CWEB program to generate solutions

Crossrefs

Cf. A003321, A046197, A052455, A052464, A124068, A124069, A226970.

Cf. A255668 (row lengths).

Programs

PARI

row(n)={m=1;while(m*9^n>=10^m,m++);for(k=1,10^m,sum(i=1,#d=digits(k),d[i]^n)==k && print1(k,", "))}
for(n=0,7,print1(row(n),", "))

Python

from itertools import combinations_with_replacement
A252648_list = [1]
for m in range(1,21):
    l, L, dm, xlist, q = 1, 1, [d**m for d in range(10)], [0], 9**m
    while l*q >= L:
        for c in combinations_with_replacement(range(1,10),l):
            n = sum(dm[d] for d in c)
            if sorted(int(d) for d in str(n)) == [0]*(len(str(n))-len(c))+list(c):
                xlist.append(n)
        l += 1
        L *= 10
    A252648_list.extend(sorted(xlist)) # Chai Wah Wu, Jan 04 2016

Extensions

I added two links. - Don Knuth, Sep 10 2015