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

0,4

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

Links

Table of n, a(n) for n=0..55.

Don Knuth, Table of a(n) for n=0..732

Don Knuth, CWEB program to generate solutions

Example

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

Prog

(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

Crossrefs

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

Cf. A255668 (row lengths).

Sequence in context: A093691 A004176 A085124 * A054054 A115353 A031298

Adjacent sequences: A252645 A252646 A252647 * A252649 A252650 A252651

Keyword

nonn,base,tabf

Author

Derek Orr, Dec 19 2014

Extensions

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

Status

approved