## Monday, March 23, 2015

### (x, why?) Mini: Narcissistic

(Click on the comic if you can't see the full image.)

I love numbers, but some numbers love themselves even more!

FYI: A Narcissistic number is any number that is equal to the sum of each of its digits raised to a power equal to the number of digits in the original number. For example, 153 = 13 + 53 + 33 and 1634 = 14 + 64 + 34 + 44.

Unlike prime numbers, it can be shown that the number of Narcissistic Numbers (also called Armstrong numbers) is finite.

Come back often for more funny math comics. Okay?

Unknown said...

You knew with a title of Narcissistic that I'd have to comment, right? :-)

Two of the four 3-digit (normal Base Ten) Narcissistic numbers are sequential, 370 & 371. Other pairs are at base ten are bigger:
24678050 & 24678051 is next, several more, and the very last two entries of the finite list of 88 are also:
115132219018763992565095597973971522400
& 115132219018763992565095597973971522401. That's a total 8 pairs and 1..9 which trivially in sequence and Narcissistic, so 27/88 are sequential. Wow.

In Base Eight, there are sequential pairs of Narcissistic N,N+1 of both length 3 and length 5, and two longer ones (that's 7+8 of 62). OEIS A010354. And big gaps.

Base Sixteen, there are two such sequences of length 3 and six of length 5 (6x5); 1x7, 1x21, 2x29, 1x40, 3x41, 1x49, 1x51. That's (15+18)/293 ~ 11%, a lower ratio.

Someone must have posed the questions:

Are there any numbers N that have Narcissistic representations in more than one Base B?

How many such (non-trivial) pairs are there in Base B total, as a function of B?
(The trivial sequence is B-1 long, for B-2 pairs; unless you count 0, then B long, B-1 pairs.)

Is there a base with 3 non-trivial Narcissistic numbers in successor sequence ?

What arithmetic progressions are possible in Narcissistic numbers for base B (besides trivial 1..B-1) ?

But i don't have time now to search if they've asked or answered those ...

Bill in Boston / n1vux

(x, why?) said...

"You knew with a title of Narcissistic that I'd have to comment, right? :-)"

I would never accuse anyone of such ... behavior. I'd let them claim that for themselves. ;)