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.
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2 comments:
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
"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. ;)
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