That matches my modular calculations from earlier, though I didn't go quite as far. If accurate it suggests asymptotically (e/4)^n n-digit examples -- still finitely many, though not quite as few as the 5/8 the simpler heuristic suggested (since e/4 < 5/8). Charles Greathouse Analyst/Programmer Case Western Reserve University On Wed, Apr 18, 2012 at 10:57 PM, Victor Miller <victorsmiller@gmail.com> wrote:
I'm doing some calculations pertaining to this. For all positive integers k define the set
S_k := { 1 <= n < 10^k | OddDigits(n) & OddDigits((n^3) mod 10^k) }
It looks like (log # S_k)/k --> 1 (where log is the natural log). Notice that 10^k-1 is always in S_k.
The sequence for #S_k starts:
[5,25,62,151,381,833,2163,5291,13317,33519,85179,213083,539212,1344272,3358571]
and the sequence (log # S_k)/k starts:
[1.6094379124341003,1.6094379124341003,1.3757114616816972,1.254319959203731,1.1885598750253403, 1.1208389403611405,1.097035917993294,1.0717203178630164,1.0551996326113493,1.0419867721215448, 1.0320463821421144,1.0224530866792414,1.0152203148865746,1.0079545114672392,1.0018017429237103]
On Wed, Apr 18, 2012 at 11:09 AM, James Buddenhagen <jbuddenh@gmail.com>wrote:
Suppose the positive integers n and n^3 both have all digits odd. Such numbers include 1, 11, 15, 33, 39, 71, 91, 173, 175, 179, 335, 3337, 5597, 7353. Is this list complete? This sequence is http://oeis.org/A085597 in OEIS. But there is no key word indicating that it is a finite sequence. Is this known to be a finite sequence?
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