Interesting Integers - Armstrong Numbers

Armstrong numbers are n digit integers that are are equal to the sum of the nth power of their digits.  For example 370 has 3 digits and 370 = 3³ + 7³ + 0³ = 27 + 343 + 0.

Other examples of Armstrong numbers are:

1,634 = 1⁴ +6⁴ + 3⁴ + 4⁴

54,748 = 5⁵ + 4⁵ + 7⁵ + 4⁵ + 8⁵

548,834 = 5⁶ +4⁶ + 8⁶ + 8⁶ + 3⁶ + 4⁶

1,741,725 = 1⁷ + 7⁷ + 4⁷ + 1⁷ + 7⁷ + 2⁷ + 5⁷

24,678,050 = 2⁸ + 4⁸ + 6⁸ + 7⁸ + 8⁸ + 0⁸ + 5⁸ + 0⁸

534,494,836 = 5⁹ + 3⁹ + 4⁹ + 4⁹ + 9⁹ + 4⁹ + 8⁹ + 3⁹ + 6⁹

4,679,307,774 = 4¹⁰ + 6¹⁰ + 7¹⁰ + 9¹⁰ + 3¹⁰ + 0¹⁰ + 7¹⁰ + 7¹⁰ + 7¹⁰ + 4¹⁰

94,204,591,914 = 9¹¹ + 4¹¹ + 2¹¹ + 0¹¹ + 4¹¹ + 5¹¹ + 9¹¹ + 1¹¹ + 9¹¹ + 1¹¹ + 4¹¹

28,116,440,335,967 = 2¹⁴ + 8¹⁴ + 1¹⁴ + 1¹⁴ + 6¹⁴ + 4¹⁴ + 4¹⁴ + 0¹⁴ + 3¹⁴ + 3¹⁴ + 5¹⁴ + 9¹⁴ + 6¹⁴ + 7¹⁴

4,338,281,769,391,370 = 4¹⁶ + 3¹⁶ + 3¹⁶ + 8¹⁶ + 2¹⁶ + 8¹⁶ + 1¹⁶ + 7¹⁶ + 6¹⁶ + 9¹⁶ + 3¹⁶ + 9¹⁶ + 1¹⁶ + 3¹⁶ + 7¹⁶ + 0¹⁶

35,875,699,062,250,035 = 3¹⁷ + 5¹⁷ + 8¹⁷ + 7¹⁷ + 5¹⁷ + 6¹⁷ + 9¹⁷ + 9¹⁷ + 0¹⁷ + 6¹⁷ + 2¹⁷ + 2¹⁷ + 5¹⁷ + 0¹⁷ + 0¹⁷ + 3¹⁷ + 5¹⁷

1,517,841,543,307,505,039 = 1¹⁹ + 5¹⁹ + 1¹⁹ + 7¹⁹ + 8¹⁹ + 4¹⁹ + 1¹⁹ + 5¹⁹ + 4¹⁹ + 3¹⁹ + 3¹⁹ + 0¹⁹ + 7¹⁹ + 5¹⁹ + 0¹⁹ + 5¹⁹ + 0¹⁹ + 3¹⁹ + 9¹⁹

63,105,425,988,599,693,916 = 6²⁰ + 3²⁰ + 1²⁰ + 0²⁰ + 5²⁰ + 4²⁰ + 2²⁰ + 5²⁰ + 9²⁰ + 8²⁰ + 8²⁰ + 5²⁰ + 9²⁰ + 9²⁰ + 6²⁰ + 9²⁰ + 3²⁰ + 9²⁰ + ²⁰ + 16²⁰

128,468,643,043,731,391,252 = 1²¹ + 2²¹ + 8²¹ + 4²¹ + 6²¹ + 8²¹ + 6²¹ + 4²¹ + 3²¹ + 0²¹ + 4²¹ + 3²¹ + 7²¹ + 3²¹ + 1²¹ + 3²¹ + 9²¹ + 1²¹ + 2²¹ + 5²¹ + 2²¹

35,452,590,104,031,691,935,943 = 3²³ + 5²³ + 4²³ + 5²³ + 2²³ + 5²³ + 9²³ + 0²³ + 1²³ + 0²³ + 4²³ + 0²³ + 3²³ + 1²³ + 6²³ + 9²³ + 1²³ + 9²³ + 3²³ + 5²³ + 9²³ + 4²³ + 3²³

174,088,005,938,065,293,023,722 = 1²⁴ + 7²⁴ + 4²⁴ + 0²⁴ + 8²⁴ + 8²⁴ + 0²⁴ + 0²⁴ + 5²⁴ + 9²⁴ + 3²⁴ + 8²⁴ + 0²⁴ + 6²⁴ + 5²⁴ + 2²⁴ + 9²⁴ + 3²⁴ + 0²⁴ + 2²⁴ + 3²⁴ + 7²⁴ + 2²⁴ + 2²⁴

4,422,095,118,095,899,619,457,938 = 4²⁵ + 4²⁵ + 2²⁵ + 2²⁵ + 0²⁵ + 9²⁵ + 5²⁵ + 1²⁵ + 1²⁵ + 8²⁵ + 0²⁵ + 9²⁵ + 5²⁵ + 8²⁵ + 9²⁵ + 9²⁵ + 6²⁵ + 1²⁵ + 9²⁵ + 4²⁵ + 5²⁵ + 7²⁵ + 9²⁵ + 3²⁵ + 8²⁵

121,204,998,563,613,372,405,438,066 = 1²⁷ + 2²⁷ + 1²⁷ + 2²⁷ + 0²⁷ + 4²⁷ + 9²⁷ + 9²⁷ + 8²⁷ + 5²⁷ + 6²⁷ + 3²⁷ + 6²⁷ + 1²⁷ + 3²⁷ + 3²⁷ + 7²⁷ + 2²⁷ + 4²⁷ + 0²⁷ + 5²⁷ + 4²⁷ + 3²⁷ + 8²⁷ + 0²⁷ + 6²⁷ + 6²⁷

23,866,716,435,523,975,980,390,369,295 = 2²⁹ + 3²⁹ + 8²⁹ + 6²⁹ + 6²⁹ + 7²⁹ + 1²⁹ + 6²⁹ + 4²⁹ + 3²⁹ + 5²⁹ + 5²⁹ + 2²⁹ + 3²⁹ + 9²⁹ + 7²⁹ + 5²⁹ + 9²⁹ + 8²⁹ + 0²⁹ + 3²⁹ + 9²⁹ + 0²⁹ + 3²⁹ + 6²⁹ + 9²⁹ + 2²⁹ + 9²⁹ + 5²⁹

1,145,037,275,765,491,025,924,292,050,346 = 1³¹ + 1³¹ + 4³¹ + 5³¹ + 0³¹ + 3³¹ + 7³¹ + 2³¹ + 7³¹ + 5³¹ + 7³¹ + 6³¹ + 5³¹ + 4³¹ + 9³¹ + 1³¹ + 0³¹ + 2³¹ + 5³¹ + 9³¹ + 2³¹ + 4³¹ + 2³¹ + 9³¹ + 2³¹ + 0³¹ + 5³¹ + 0³¹ + 3³¹ + 4³¹ + 6³¹

17,333,509,997,782,249,308,725,103,962,772 = 1³² + 7³² + 3³² + 3³² + 3³² + 5³² + 0³² + 9³² + 9³² + 9³² + 7³² + 7³² + 8³² + 2³² + 2³² + 4³² + 9³² + 3³² + 0³² + 8³² + 7³² + 2³² + 5³² + 1³² + 0³² + 3³² + 9³² + 6³² + 2³² + 7³² + 7³² + 2³²

186,709,961,001,538,790,100,634,132,976,990 = 1³³ + 8³³ + 6³³ + 7³³ + 0³³ + 9³³ + 9³³ + 6³³ + 1³³ + 0³³ + 0³³ + 1³³ + 5³³ + 3³³ + 8³³ + 7³³ + 9³³ + 0³³ + 1³³ + 0³³ + 0³³ + 6³³ + 3³³ + 4³³ + 1³³ + 3³³ + 2³³ + 9³³ + 7³³ + 6³³ + 9³³ + 9³³ + 0³³

1,122,763,285,329,372,541,592,822,900,204,593 = 1³⁴ + 1³⁴ + 2³⁴ + 2³⁴ + 7³⁴ + 6³⁴ + 3³⁴ + 2³⁴ + 8³⁴ + 5³⁴ + 3³⁴ + 2³⁴ + 9³⁴ + 3³⁴ + 7³⁴ + 2³⁴ + 5³⁴ + 4³⁴ + 1³⁴ + 5³⁴ + 9³⁴ + 2³⁴ + 8³⁴ + 2³⁴ + 2³⁴ + 9³⁴ + 0³⁴ + 0³⁴ + 2³⁴ + 0³⁴ + 4³⁴ + 5³⁴ + 9³⁴ + 3³⁴

12,639,369,517,103,790,328,947,807,201,478,392 = 1³⁵ + 2³⁵ + 6³⁵ + 3³⁵ + 9³⁵ + 3³⁵ + 6³⁵ + 9³⁵ + 5³⁵ + 1³⁵ + 7³⁵ + 1³⁵ + 0³⁵ + 3³⁵ + 7³⁵ + 9³⁵ + 0³⁵ + 3³⁵ + 2³⁵ + 8³⁵ + 9³⁵ + 4³⁵ + 7³⁵ + 8³⁵ + 0³⁵ + 7³⁵ + 2³⁵ + 0³⁵ + 1³⁵ + 4³⁵ + 7³⁵ + 8³⁵ + 3³⁵ + 9³⁵ + 2³⁵

1,219,167,219,625,434,121,569,735,803,609,966,019 = 1³⁷ + 2³⁷ + 1³⁷ + 9³⁷ + 1³⁷ + 6³⁷ + 7³⁷ + 2³⁷ + 1³⁷ + 9³⁷ + 6³⁷ + 2³⁷ + 5³⁷ + 4³⁷ + 3³⁷ + 4³⁷ + 1³⁷ + 2³⁷ + 1³⁷ + 5³⁷ + 6³⁷ + 9³⁷ + 7³⁷ + 3³⁷ + 5³⁷ + 8³⁷ + 0³⁷ + 3³⁷ + 6³⁷ + 0³⁷ + 9³⁷ + 9³⁷ + 6³⁷ + 6³⁷ + 0³⁷ + 1³⁷ + 9³⁷

12,815,792,078,366,059,955,099,770,545,296,129,367 = 1³⁸ + 2³⁸ + 8³⁸ + 1³⁸ + 5³⁸ + 7³⁸ + 9³⁸ + 2³⁸ + 0³⁸ + 7³⁸ + 8³⁸ + 3³⁸ + 6³⁸ + 6³⁸ + 0³⁸ + 5³⁸ + 9³⁸ + 9³⁸ + 5³⁸ + 5³⁸ + 0³⁸ + 9³⁸ + 9³⁸ + 7³⁸ + 7³⁸ + 0³⁸ + 5³⁸ + 4³⁸ + 5³⁸ + 2³⁸ + 9³⁸ + 6³⁸ + 1³⁸ + 2³⁸ + 9³⁸ + 3³⁸ + 6³⁸ + 7³⁸

115,132,219,018,763,992,565,095,597,973,971,522,401 = 1³⁹ + 1³⁹ + 5³⁹ + 1³⁹ + 3³⁹ + 2³⁹ + 2³⁹ + 1³⁹ + 9³⁹ + 0³⁹ + 1³⁹ + 8³⁹ + 7³⁹ + 6³⁹ + 3³⁹ + 9³⁹ + 9³⁹ + 2³⁹ + 5³⁹ + 6³⁹ + 5³⁹ + 0³⁹ + 9³⁹ + 5³⁹ + 5³⁹ + 9³⁹ + 7³⁹ + 9³⁹ + 7³⁹ + 3³⁹ + 9³⁹ + 7³⁹ + 1³⁹ + 5³⁹ + 2³⁹ + 2³⁹ + 4³⁹ + 0³⁹ + 1³⁹

The number above is the largest Armstrong number.  It has been proven that there are only 88 Armstrong numbers.

The number below is larger, but it is not an Armstrong number.  Can you tell me why?  I have included it here because it is still an interesting integer - even if it is not an Armstrong number.

36,428,594,490,313,158,783,584,452,532,870,892,261,556 = 3⁴² + 6⁴² + 4⁴² + 2⁴² + 8⁴² + 5⁴² + 9⁴² + 4⁴² + 4⁴² + 9⁴² + 0⁴² + 3⁴² + 1⁴² + 3⁴² + 1⁴² + 5⁴² + 8⁴² + 7⁴² + 8⁴² + 3⁴² + 5⁴² + 8⁴² + 4⁴² + 4⁴² + 5⁴² + 2⁴² + 5⁴² + 3⁴² + 2⁴² + 8⁴² + 7⁴² + 0⁴² + 8⁴² + 9⁴² + 2⁴² + 2⁴² + 6⁴² + 1⁴² + 5⁴² + 5⁴² + 6⁴²

REFERENCES:

http://oeis.org/A005188

http://www.futilitycloset.com/2011/07/23/math-notes-91/

David