

A261433


kdigit integers equal to the sum of the kth powers of the tens' complements of their digits.


3




OFFSET

1,1


COMMENTS

The terms of the sequence could be called "Shy n ndigit numbers" as suggested by Geoffrey Campbell, Long Term Visitor (Visiting Fellow), Mathematical Sciences Institute, Australian National University, cf Links.
In base 10, x is a "Shy k ndigit number" if it is an ndigit (d_i) number such that x = Sum_{i=1..n}{(10d_i)^k}. For instance, 2240 is a "Shy 3 4digit number": (10  2)^3 + (10  2)^3 + (10  4)^3 + (10  0)^3 = 512 + 512 + 216 + 1000 = 2240. Again, 2149042 is a "Shy 6 7digit 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.
It is not known if the sequence is finite. At least there are no other terms up to 18digit numbers (as tested by Marco Cecchi at LinkedIn link).
If there are further terms, they are greater than 10^33.  Giovanni Resta, Aug 20 2015
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^(k1) (i.e., have fewer than k digits) if k > 60.  Chai Wah Wu, Apr 07 2018


LINKS



EXAMPLE

(10  5)^1 = 5,
(10  3)^3 + (10  7)^3 + (10  8)^3 = 343 + 27 + 8 = 378,
(10  9)^5 + (10  1)^5 + (10  8)^5 + (10  8)^5 + (10  2)^5 = 1 + 59049 + 32 + 32 + 32768 = 91882, etc.


MAPLE

with(numtheory): P:=proc(q) local a, b, c, k, n;
for n from 1 to q do a:=ilog10(n)+1; b:=0; c:=n;
for k from 1 to a do b:=b+(10(c mod 10))^a; c:=trunc(c/10); od;
if b=n then print(n); fi; od; end: P(10^9);


MATHEMATICA

Select[Range[10^5], # == Total[(10  IntegerDigits@ #)^ IntegerLength[#]] &] (* Giovanni Resta, Aug 20 2015 *)


PROG

(PARI) isok(n) = (d = digits(n)) && (sum(k=1, #d, (10d[k])^#d) == n); \\ Michel Marcus, Aug 24 2015
(Python)
from itertools import combinations_with_replacement
for k in range(1, 10):
a, k10 = tuple([i**k for i in range(10, 0, 1)]), 10**k
for b in combinations_with_replacement(range(1, 10), k):
x = sum(list(map(lambda y:a[y], b)))
if x < k10 and tuple(int(d) for d in sorted(str(x))) == b:


CROSSREFS



KEYWORD

nonn,base,fini,full


AUTHOR



EXTENSIONS

a(4)a(6) found by Aleksander Zujev


STATUS

approved



