Self Power Numbers - Unfinished business!

Sat Feb 2 20:06:46 CET 2008

This raises the question of which bases have self power numbers that
do not include digits 0 or 1.  Obviously, 2 does not, since it has no 
other
digits.  Base 3 also does not; any number all 2's is of the form 3^n - 
1,
which is never a square for n > 0.  (When n is even, 3^n is a square, 
so
3^n - 1 cannot be; when n is odd, 3^n - 1 = 2 (mod 4), which cannot
be a square.)

In base 4, any square ends in 0 or 1, so the number would have to
consist of all 3's.  A similar argument to that above shows that no
number 4^n - 1 can be a cube,  looking at the numbers modulo 9.

We can define a complete self power number as a self power number
that includes all digits except 0 and 1, but does not include 0 or 1.
The question of which bases have complete self power numbers is a
bit more approachable.  In particular, no prime power base does: if
n = p^k, any phi(p^k)-th power (= (p-1)*p^{k-1}) ends in 0 or 1.

It may be that every other base does have a complete self power
number.  Generally, polynomials grow slowly enough that one can avoid
a digit or two, as long as there are no modular problems.  Base 6 does
have a complete self power number: 5243 (1179 in decimal).

This argument lets us (almost) show that there are no non-trivial self
power numbers in base 5.  Such a number could not include a digit 4.
Squares in base 5 must end in 0, 1, or 4, so we must exclude digit 2
as well; this leaves only 3 as an allowable digit.  To complete the 
proof,
we need to show that numbers of the form 3*(5^n-1)/4 cannot be
cubes.  Looking mod 27 shows that n would have to be a multiple of
6.  I am almost certain that there are no such cubes; certainly there
are none up to n = 120.

Franklin T. Adams-Watters

-----Original Message-----
From: N. J. A. Sloane <njas at research.att.com>

Dear Seqfans,  (The OEIS machine is still down)

Back in 2005 there was a discussion which began like this (I've edited
it a bit):

From: "Eric Angelini" <keynews.tv at skynet.be>
Date: Tue, 5 Jul 2005 15:50:24 +0200
Subject: Self-powers numbers (SPN)

The number N=325648 might be called a self-power
number because:

"3" means "a cube is visible in N"
    (yes, it is "8" -- 8=2*2*2)
"2" means "a square is visible in N"
    (yes, it is "25" -- 25=5*5)
    (4 is ok too, being 2*2)
"5" means "a power 5 is visible in N"
    (yes, it is "32" = 2*2*2*2*2)
"6" means "a power 6 is visible in N"
    (yes, it is "64" = 2*2*2*2*2*2)
"4" means "a power 4 is visible in N"
    (yes, it is "256" = 4*4*4*4*)
"8" means "a power 8 is visible in N"
    (yes, it is "256" =2*2*2*2*2*2*2*2)

N=832564 is a SPN too, of course.

["visible" means "as a whole": "25" is
NOT visible in 235]

Question:
Can someone compute all such SPN _which
don't include any 0's or 1's_ ?

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Thanks to everyone who responded.  The following will
be visible next time the OEIS machine is working.
Neil

%I A134698
%S A134698 32564,232564,256432,322564,325642,325643,325644,325645,325646,
%T A134698 325648,325664,332564,432564,532564,632564,643256,832564
%N A134698 Self-power numbers (or SPN's): numbers n with property that for each digit d of n, we can also see the decimal expansion of the d-th power of some number as a substring of n, and also n contains no 0's or 1's.
%C A134698 a(1) - a(17) computed by Maximilian Hasler (maximilian.hasler(AT)gmail.com), Feb 02 2008
%C A134698 There are an infinite number of SPN's, since for example we can prefix n by any digit of n.
%e A134698 n = 32564 is a member because we can see a cube (64) in n, a square (4 or 25), a fifth power (32), a sixth power (64) and a fourth power (256).
%O A134698 1,1
%K A134698 nonn,base,more
%Y A134698 Cf. A134947, A134439, A134692, A134948.
%o A134698 (PARI code from Maximilian Hasler) isSPN(n)={ local( V=Vec(Str(n)), S=eval(Set(V)), t); S[1]<2 & return; /* disallow 0 ot 1*/
%o A134698 t=eval( Set( concat( vector( #V,a,vector( a,b,concat( vecextract( V,2^a-2^(b-1))))))));
%o A134698 for(i=1,#S, for(j=1,#t, floor(sqrtn(t[j]+.5,S[i]))^S[i]==t[j] & next(2) /*OK: goto next power*/);
%o A134698 return /*not OK*/); n}
%o A134698 for(n=1,10^9, isSPN(n) |next; print1(n","))
%A A134698 Eric Angelini (keynews.tv(AT)skynet.be), Jul 05 2005

%I A134947
%S A134947 32564,322564,325648,325664,643256
%N A134947 Primitive self-power numbers (or SPN's): terms in A134698 which cannot be obtained by taking an earlier term M (say) of A134698 and prefixing or suffixing a digit of M.
%C A134947 a(1) - a(5) computed by Maximilian Hasler (maximilian.hasler(AT)gmail.com), Feb 02 2008
%C A134947 One could also consider a stronger notion of primitivity, which would exclude numbers that can be obtained from an SPN by inserting a duplicated digit anywhere in the number. - Maximilian Hasler. This sequence, which would begin 32564, ..., is not yet in the OEIS.
%e A134947 232564 is an SPN but is not primitive since it can be obtained from the SPN 32564 by prefixing the digit 2.
%O A134947 1,1
%K A134947 nonn,base,more
%Y A134947 Cf. A134698.
%A A134947 Eric Angelini (keynews.tv(AT)skynet.be), Jul 05 2005

%I A134948
%S A134948 1,2,10,11,12,20,21,22,24,100,101,102,110,111,112,120,121,122,124
%N A134948 Self-factorial numbers: numbers n with property that for each digit d of n, we can also see the decimal expansion of d! as a substring of n.
%e A134948 24 is a self-factorial number because we can see both 2! = 2 and 4! = 24 in the decimal exapnsion 24.
%O A134948 1,2
%K A134948 nonn,base,more
%Y A134948 Cf. A134698, A134947, A134439, A134692.
%A A134948 Alexander Povolotsky (apovolot(AT)gmail.com), Feb 02 2008
%E A134948 a(1) - a(19) computed by njas, Feb 02 2008

%I A134439
%S A134439 2536497816,2536498167,2573649816,2578163649,2581636497,2581673649,
%T A134439 3649257816,3649258167,3649725816,3649781625,3649816257,3649816725,
%U A134439 7253649816,7258163649,7364925816,7364981625,7816253649,7816364925,8162536497,8162573649,8163649257,8163649725,8167253649,8167364925
%N A134439 Numbers n with property that for each digit of n, we can also see the decimal expansion of d^2 as a substring of n.
%C A134439 This list shown includes all such numbers with <= 10 digits.
%e A134439 In the first number, for 2 we can see 4, for 5 we can see 25, for 3 we can see 9, for 6 we can see 36, for 4 we can see 16, for 9 we can see 81, for 8 we can see 64, for 1 we can see 1, for 7 we can see 49.
%O A134439 1,1
%K A134439 nonn,base
%Y A134439 Cf. A134692, A134698.
%A A134439 Zak Seidov (zakseidov(AT)yahoo.com),Feb 02 2008

%I A134692
%S A134692 1251272921643438,1251272921648343,1251272934321648,1251272934382164,
%T A134692 1251272982164343,1251272983432164,2164125127293438,2164125127298343,
%U A134692 2164343125127298,2164343812512729,2164812512729343,2164834312512729
%N A134692 Numbers n with property that for each digit of n, we can also see the decimal expansion of d^3 as a substring of n.
%C A134692 See A134692 for further information.
%O A134692 1,1
%H A134692 Zak Seidov, <a href="http://www.research.att.com/~njas/sequences/b134692.txt">Table of n, a(n) for n = 1..24</a>
%K A134692 nonn,base
%Y A134692 Cf. A134439, A134698.
%A A134692 Zak Seidov (zakseidov(AT)yahoo.com),Feb 02 2008

Not specifically, but in a statistics class at Rice we once computed  
the expected number of moves required by a knight on a central square  
of a chessboard to return to its starting square, making legal moves  
at random. Surprise: the answer is 42. One could generalize that  
problem as well, but the Tower of Hanoi problem seems much more natural.

Bob

On Feb 1, 2008, at 8:32 PM, N. J. A. Sloane wrote:

> My former colleague Toby Berger (now at U of Va)
> has been looking at this problem.  Has anyone seen
> anything like this before?
>
> %I A134939
> %S A134939 0,2,64,1274,21760
> %N A134939 Consider a 3-pole Tower of Hanoi configuration which  
> begins with n rings on pole 1. Moves are made at random, where the 1- 
> step transition probabilities out of any state are equal. Let e(n)  
> be the expected number of transitions to reach the state in which  
> which all rings are on pole 3. Sequence gives a(n), the numerator of  
> e(n).
> %C A134939 Both allowable transitions out of any of the three  
> special states in which all the rings are on one of the poles have  
> probabilty 1/2, and each of the three allowable transitions out of  
> any of the other 3^n - 3 states have probabilty 1/3.
> %C A134939 It appears that the denominator of e(n) for n>=1 is  
> 3^(n-1).
> %e A134939 The values of e(0), ..., e(4) are 0, 2, 64/3, 1274/9,  
> 21760/27.
> %K A134939 nonn,frac,more,new
> %Y A134939 Cf. A134940.
> %O A134939 0,2
> %A A134939 Toby Berger (tb6n(AT)virginia.edu), Jan 23 2008
>
> Neil
>

I've examined some upper bounds for the number of grid "anchor" points in a 
finite (n+1) by (n+1) square grid that define C(a(n),2) lines which
cover all of the (n+1)^2 grid points as defined in A116446. The results
are in http://www.strw.leidenuniv.nl/~mathar/progs/a116446.pdf .

Richard