[seqfan] Re: Sloane's Sequence A023052

[Hans Havermann]

Robert G. Wilson, v:

> Dear Sir,
>
>    See http://mathworld.wolfram.com/NarcissisticNumber.html

>> The short answer is no, we do not know whether A023052 is finite.

I'm tempted to embrace the (incorrect) definition of Harvey Heinz <http://www.geocities.com/CapeCanaveral/Launchpad/4057/narciss.htm#PDIs 
 > that a "perfect digital invariant" (PDI) "is a number equal to the  
sum of a power of its digits when the power is NOT equal to the length  
of the number". Unfortunately, Joseph Madachy's original definition  
(at least as reprinted in his "Mathematical Recreations", Dover 1979)  
appears to INCLUDE numbers where the power is equal to the length of  
the digits, although it EXCLUDES both zero and one. In the linked  
MathWorld article, a statement that "as summarized in the table above,  
a total of 88 narcissistic numbers exist in base 10" obviously  
excludes zero (there are 89 entries in the table) but not one. Our  
very own A003321 (smallest n-th order perfect digital invariant)  
begins "2, 0, 153, ...", the greater-than-one having been incorporated  
into the definition. You get the idea: The historical concept of  
narcissistic numbers and digital invariants still struggles to find a  
champion who will drag the definitional morass into a comprehensive  
standard.

But let me get back to Harvey Heinz's "PDI". If we exclude from  
A023052 ("powerful" numbers: yet another confusion) numbers where the  
power is equal to the length of the number, what remains?

  1                                      4150 (+1)
  2                                      4151 (+1)
  3                                    194979 (-1)
  4                                  14459929 (-1)
  5                              564240140138 (+1)
  6                           233411150132317 (+2)

The bracketed plus or minus gives the power in relation to the length  
of the number. Can we extend this? I had previously noted that Richard  
Mathar's supplied link <http://bbs.emath.ac.cn/viewthread.php?action=printable&tid=1104 
 > might not be exhaustive because I did not find either 194979 or  
14459929 thereon. That caveat notwithstanding, the page does sport  
another twenty candidates:

  7                  114735624485461118832514 (+1)
  8                  832662335985815242605070 (+1)
  9                  832662335985815242605071 (+1)
10                 7584178683470015004720746 (+2)
11                77888878776432530886487094 (+1)
12               477144170826130800418527003 (+2)
13              4716716265341543230394614213 (+1)
14              5022908050052864745436221003 (+1)
15            793545620525277858657607629822 (+1)
16          32186410459473623435614002227248 (+1)
17        5250083909873201044638631458484846 (+1)
18        7673249664848285722449710136138169 (+1)
19       91097771122214850683543503173498149 (+1)
20      418510620208153136884574959404115822 (+1)
21      618670315011216949642642321868915308 (+1)
22     7320233109580046612992702336326619665 (+1)
23     7403697806790834730831423191927508283 (+1)
24    16427762135335641330720936105651700735 (+1)
25    83281823928125880164896079973522049472 (+1)
26    83281830613691836766959173718984508549 (+1)

Finally, we have Lionel E. Deimel, Jr. and Michael T. Jones with their  
1982 discovery:

27 36428594490313158783584452532870892261556 (+1)

By the way, Deimel and Jones state this about their find: "Although  
proving nothing, it lends support to our conjecture that all bases  
greater than 2 have an infinite number of PDIs." Although I have  
indexed all of these entries, it should be understood that the  
veracity of that index depends on the completeness of entries #7 to  
#26, which I have no way of verifying.

Anyone who wishes to create a (provisional) b-file for A023052 can now  
collate my list (barring errors) with the classic (finite)  
narcissistic numbers (A005188): Add zero, if you like. It is  
unfortunate that at present the extensions to 10^50 by G.N. Gusev and  
to 10^74 by Xiaoqing Tang are little more than unverifiable footnotes.