perplexus.info :: Numbers : Armstrong plus one : Armstwo

Armstrong plus one : Armstwo (Posted on 2021-12-27)

An Armstrong number is a positive integer that equals the sum of M-th powers of their digits when the number is M-digit long.
153 is an Armstrong number, since: 1³+5³+3³=153.
Sloane's A005188 has an article on this, in which inter-alia it is mentioned that the sequence of Armstrong numbers terminates at the 88th term.

An Armstwo number is a base ten, M-digit long positive integer which is equal to the sum of M-th powers of one greater than each of the digits.
For example, if we check for 153, we find that:
2³+6³+4³= 288, which is NOT equal to 153.

Determine the smallest Armstwo number.

**** Heartfelt thanks to Larry for inspiring this puzzle.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

| Comment 1 of 3

A sincere "you're welcome" and many thanks to K Sengupta for inspirational credit.

I tested integers up to 10^7 and only found three solutions, the smallest being 141.

{141, 251, 560} is the set of solutions I found.

here is my Python code (I included a function to test for Armstrong numbers as a warm-up to testing for Armstwo numbers)

--------------

def isArmstrong(n):

digits = list(str(n))

power = len(digits)

ans = 0

for d in digits:

ans += int(d)**power

return ans == n

def isArmstwo(n):

digits = list(str(n))

power = len(digits)

ans = 0

for d in digits:

ans += (int(d)+1)**power

return ans == n

big = 10000000

count = 0

for i in range(big):

if isArmstwo(i):

print(i, 'is an Armstwo number')

count += 1

print(count)

--------------

Output:

141 is an Armstwo number

251 is an Armstwo number

560 is an Armstwo number

Posted by Larry on 2021-12-27 10:35:31

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