Perfect digit-to-digit invariant

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A perfect digit-to-digit invariant (PDDI) (also known as a Canouchi number[1]) is a number that is equal to the sum of its digits each raised to a power equal to the digit.

n = d_k^{d_k} + d_{k-1}^{d_{k-1}} + \dots + d_2^{d_2} + d_1^{d_1}\,.

1 is PDDI in any base. Apart from 1 there is only one other PDDI in the decimal system, 3435. It is important to note that 0 and 438579088 (sequence A046253 in OEIS), are PDDIs under the convention that 00 = 0, but this is not standard usage.[2][3]

3^3 + 4^4 + 3^3 + 5^5 = 27 + 256 + 27 + 3125 = 3435
4^4 + 3^3 + 8^8 + 5^5 + 7^7 + 9^9 + 0^0 + 8^8 + 8^8
= 256 + 27 + 16777216 + 3125 + 823543 + 387420489 + 0 + 16777216 + 16777216 = 438579088

More generally, there are finitely many PDDIs in any base.

References[edit]

  1. ^ van Berkel, Daan (2009). "On a curious property of 3435". arXiv:0911.3038 [math.HO].
  2. ^ Narcisstic Number, Harvey Heinz
  3. ^ Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin. p. 185. ISBN 0-14-026149-4. 

External links[edit]