Perfect digit-to-digit invariant


Perfect digit-to-digit invariant

A perfect digit-to-digit invariant (PDDI) (also known as a Munchausen 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}\,.

0 and 1 are PDDIs in any base (using the convention that 00 = 0). Apart from 0 and 1 there are only two other PDDIs in the decimal system, 3435 and 438579088 (sequence A046253 in OEIS).

33 + 44 + 33 + 55 = 27 + 256 + 27 + 3125 = 3435


44 + 33 + 88 + 55 + 77 + 99 + 00 + 88 + 88
= 256 + 27 + 16777216 + 3125 + 823543 + 387420489 + 0 + 16777216 + 16777216 = 438579088


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

References

  1. ^ van Berkel, Daan (2009). "On a curious property of 3435". arXiv:0911.3038. 

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