[seqfan] Re: Sloane's Sequence A023052
Hans Havermann
pxp at rogers.com
Wed Jun 17 18:06:35 CEST 2009
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.
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