cp's OEIS Frontend

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, ...

%I A208130 #47 Jan 06 2025 21:44:41
%S A208130 1,2,3,4,5,6,7,8,9,89,135,2537,60409,4901732,17735872,45279768,
%T A208130 393470463,3623008669,3893095238,229386834955666,1892713761283624,
%U A208130 1501212693940707502,1517944702855898904,12303679765763687463,122947811178635339597,1095354314191826124704,1106509957063490820877
%N 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, ...
%C A208130 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
%H A208130 Francis J. McDonnell, <a href="/A208130/a208130.txt">Java program</a>
%e A208130 2537 = 2^1 + 3^2 + 5^3 + 7^4 = 2 + 9 + 125 + 2401.
%e A208130 60409 = 0^1 + 0^2 + 4^3 + 6^4 + 9^5 = 0 + 0 + 64 + 1296 + 59049.
%o A208130 (Java) // See McDonnell link.
%o A208130 (Python)
%o A208130 from itertools import combinations_with_replacement
%o A208130 A208130_list = []
%o A208130 for l in range(1,23):
%o A208130     for n in combinations_with_replacement(range(10),l):
%o A208130         x = sum(b**(a+1) for a,b in enumerate(n))
%o A208130         if x > 0 and tuple(sorted(int(d) for d in str(x))) == n:
%o A208130             A208130_list.append(x)
%o A208130 A208130_list = sorted(A208130_list)  # _Chai Wah Wu_, May 20 2017
%Y A208130 Cf. A032799 (does not sort the digits prior to raising to powers).
%K A208130 nonn,base,fini,full
%O A208130 1,2
%A A208130 _Francis J. McDonnell_, Mar 29 2012
%E A208130 More terms added by _Francis J. McDonnell_, Apr 12 2012
%E A208130 Faster program used to obtain more terms included by _Francis J. McDonnell_, Apr 16 2012