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

5, 378, 91882, 3762938, 46478818, 564426414

Offset: 1

Author

Paolo P. Lava, Aug 20 2015

Keywords

• nonn
• base
• fini
• full

Comments

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.

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.

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).

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^(k-1) (i.e., have fewer than k digits) if k > 60. - Chai Wah Wu, Apr 07 2018

Examples

			(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.
		

Links

• Geoffrey Campbell, Related to Narcissistic numbers: the Shy numbers, Number Theory group on LinkedIn.com
• Marco Cecchi, Python program based on partitions.

Crossrefs

Cf. A005188, A052382.

Programs

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 *)

PARI

isok(n) = (d = digits(n)) && (sum(k=1, #d, (10-d[k])^#d) == n); \\ Michel Marcus, Aug 24 2015

Python

from itertools import combinations_with_replacement
A261433_list = []
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:
            A261433_list.append(x)
A261433_list = sorted(A261433_list) # Chai Wah Wu, Aug 25 2015, updated Apr 06, 2018

Extensions

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