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A208130
Numbers that when expressed in decimal are equal to the sum of the digits sorted into nondecreasing order and raised to the powers 1, 2, 3, ...
3
1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 2537, 60409, 4901732, 17735872, 45279768, 393470463, 3623008669, 3893095238, 229386834955666, 1892713761283624, 1501212693940707502, 1517944702855898904, 12303679765763687463, 122947811178635339597, 1095354314191826124704, 1106509957063490820877
OFFSET
1,2
COMMENTS
Lemma: The sequence is finite with all terms in the sequence having at most 22 digits. Proof: Let n be an m-digit natural number in the sequence for some m. Then 10^(m-1) <= n and n <= 9 + 9^2 + ... + 9^m = 9(9^m-1)/8 < (9^(m+1))/8. Thus 10^(m-1) < (9^(m+1))/8. Taking logarithms of both sides and solving yields m < 22.97. QED. The sequence listed, found by a computer program searching up to 10^22, is therefore complete. - Francis J. McDonnell, Apr 12 2012
LINKS
EXAMPLE
2537 = 2^1 + 3^2 + 5^3 + 7^4 = 2 + 9 + 125 + 2401.
60409 = 0^1 + 0^2 + 4^3 + 6^4 + 9^5 = 0 + 0 + 64 + 1296 + 59049.
PROG
(Java) see link.
(Python)
from itertools import combinations_with_replacement
A208130_list = []
for l in range(1, 23):
for n in combinations_with_replacement(range(10), l):
x = sum(b**(a+1) for a, b in enumerate(n))
if x > 0 and tuple(sorted(int(d) for d in str(x))) == n:
A208130_list.append(x)
A208130_list = sorted(A208130_list) # Chai Wah Wu, May 20 2017
CROSSREFS
Cf. A032799 (does not sort the digits prior to raising to powers).
Sequence in context: A228326 A098766 A032799 * A160343 A250265 A239085
KEYWORD
nonn,base,fini,full
AUTHOR
EXTENSIONS
More terms added by Francis J. McDonnell, Apr 12 2012
Faster program used to obtain more terms included by Francis J. McDonnell, Apr 16 2012
STATUS
approved