cp's OEIS Frontend

A261433 k-digit integers equal to the sum of the k-th powers of the tens' complements of their digits.

%I A261433 #41 Jul 10 2018 10:15:50
%S A261433 5,378,91882,3762938,46478818,564426414
%N A261433 k-digit integers equal to the sum of the k-th powers of the tens' complements of their digits.
%C A261433 The terms of the sequence could be called "Shy n n-digit numbers" as suggested by Geoffrey Campbell, Long Term Visitor (Visiting Fellow), Mathematical Sciences Institute, Australian National University, cf Links.
%C A261433 In base 10, x is a "Shy k n-digit number" if it is an n-digit (d_i) number such that x = Sum_{i=1..n}{(10-d_i)^k}. For instance, 2240 is a "Shy 3 4-digit number": (10 - 2)^3 + (10 - 2)^3 + (10 - 4)^3 + (10 - 0)^3 = 512 + 512 + 216 + 1000 = 2240. Again, 2149042 is a "Shy 6 7-digit number": (10 - 2)^6 + (10 - 1)^6 + (10 - 4)^6 + (10 - 9)^6 + (10 - 0)^6 + (10 - 4)^6 + (10 - 2)^6 =  262144 + 531441 + 46656 + 1 + 1000000 + 46656 + 262144 = 2149042.
%C A261433 It is not known if the sequence is finite. At least there are no other terms up to 18-digit numbers (as tested by Marco Cecchi at LinkedIn link).
%C A261433 If there are further terms, they are greater than 10^33. - _Giovanni Resta_, Aug 20 2015
%C A261433 Subsequence of A052382. Sequence is finite and complete as verified by exhaustive search since all terms have 60 or fewer digits. Since all terms are zeroless, they are less than k*9^k which would be less than 10^(k-1) (i.e., have fewer than k digits) if k > 60. - _Chai Wah Wu_, Apr 07 2018
%H A261433 Geoffrey Campbell, <a href="https://www.linkedin.com/grp/post/4510047-6038963758880010243"> Related to Narcissistic numbers: the Shy numbers</a>, Number Theory group on LinkedIn.com
%H A261433 Marco Cecchi, <a href="/A261433/a261433.txt">Python program based on partitions</a>.
%e A261433 (10 - 5)^1 = 5,
%e A261433 (10 - 3)^3 + (10 - 7)^3 + (10 - 8)^3 = 343 + 27 + 8 = 378,
%e A261433 (10 - 9)^5 + (10 - 1)^5 + (10 - 8)^5 + (10 - 8)^5 + (10 - 2)^5 = 1 + 59049 + 32 + 32 + 32768 = 91882, etc.
%p A261433 with(numtheory): P:=proc(q) local a,b,c,k,n;
%p A261433 for n from 1 to q do a:=ilog10(n)+1; b:=0; c:=n;
%p A261433 for k from 1 to a do b:=b+(10-(c mod 10))^a; c:=trunc(c/10); od;
%p A261433 if b=n then print(n); fi; od; end: P(10^9);
%t A261433 Select[Range[10^5], # == Total[(10 - IntegerDigits@ #)^ IntegerLength[#]] &] (* _Giovanni Resta_, Aug 20 2015 *)
%o A261433 (PARI) isok(n) = (d = digits(n)) && (sum(k=1, #d, (10-d[k])^#d) == n); \\ _Michel Marcus_, Aug 24 2015
%o A261433 (Python)
%o A261433 from itertools import combinations_with_replacement
%o A261433 A261433_list = []
%o A261433 for k in range(1,10):
%o A261433     a, k10 = tuple([i**k for i in range(10,0,-1)]), 10**k
%o A261433     for b in combinations_with_replacement(range(1,10),k):
%o A261433         x = sum(list(map(lambda y:a[y],b)))
%o A261433         if x < k10 and tuple(int(d) for d in sorted(str(x))) == b:
%o A261433             A261433_list.append(x)
%o A261433 A261433_list = sorted(A261433_list) # _Chai Wah Wu_, Aug 25 2015, updated Apr 06, 2018
%Y A261433 Cf. A005188, A052382.
%K A261433 nonn,base,fini,full
%O A261433 1,1
%A A261433 _Paolo P. Lava_, Aug 20 2015
%E A261433 a(4)-a(6) found by Aleksander Zujev