The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A186437

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A186437

Maximal number of squarings in an evaluation scheme for x^n achieving the minimal number of operations.
(history; published version)

#13 by N. J. A. Sloane at Wed Feb 04 23:25:23 EST 2015

STATUS

proposed

approved

#12 by Jon E. Schoenfield at Wed Feb 04 19:14:45 EST 2015

STATUS

editing

proposed

#11 by Jon E. Schoenfield at Wed Feb 04 19:14:34 EST 2015

EXAMPLE

x^2 = (x)^2 ; x^3 = x * x^2 ; x^5 = x^2 * x^3

x^2 = (x)^2 ; x^4 = (x^2)^2 ; x^5 = x * x^4

Discussion

Wed Feb 04

19:14

Jon E. Schoenfield: Since the Style Sheet explicitly says
     exponentiation: use ^ rather than **, ², or ³
... I figured the superscript 2 characters should go away, but I wasn't sure what to use in their place.  'Any suggestions?

#10 by Jon E. Schoenfield at Wed Feb 04 19:12:04 EST 2015

EXAMPLE

x^2 = (x² )^2 ; x^3 = x * x^2 ; x^5 = x^2 * x^3

x^2 = (x² )^2 ; x^4 = (x^2)² ^2 ; x^5 = x * x^4

STATUS

approved

editing

#9 by Russ Cox at Sat Mar 31 10:25:33 EDT 2012

AUTHOR

Laurent Thévenoux and _Christophe Mouilleron (christophe.mouilleron(AT)ens-lyon.org), _, Feb 23 2011

Discussion

Sat Mar 31

10:25

OEIS Server: https://oeis.org/edit/global/467

#8 by N. J. A. Sloane at Thu Feb 24 01:07:32 EST 2011

STATUS

proposed

approved

#7 by Christophe Mouilleron at Wed Feb 23 11:05:40 EST 2011

NAME

allocated Maximal number of squarings in an evaluation scheme for Christophe Mouilleronx^n achieving the minimal number of operations.

DATA

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

OFFSET

1,4

COMMENTS

a(n) is also the maximal number of doublings in a shortest addition chain for n.

FORMULA

We have a(n) = floor(log_2(n)) for all n ≤ 60 except 23, 39, 43 and 46.

EXAMPLE

For n=5, we can evaluate x^5 using only 3 operations in 2 ways:

x^2 = x² ; x^3 = x * x^2 ; x^5 = x^2 * x^3

x^2 = x² ; x^4 = (x^2)² ; x^5 = x * x^4

The second way achieves the maximal number of doublings, which is a(5) = 2.

CROSSREFS

Cf A003313.

KEYWORD

allocated

nonn

AUTHOR

Laurent Thévenoux and Christophe Mouilleron (christophe.mouilleron(AT)ens-lyon.org), Feb 23 2011

STATUS

approved

proposed

#6 by Christophe Mouilleron at Wed Feb 23 11:05:40 EST 2011

NAME

allocated for Christophe Mouilleron

KEYWORD

recycled

allocated

#5 by Charles R Greathouse IV at Tue Feb 22 20:49:02 EST 2011

STATUS

proposed

approved

#4 by Charles R Greathouse IV at Tue Feb 22 20:48:53 EST 2011

NAME

Numbers such that the sum of the digits plus the product of the digits equals the square of the sum of the digits.

DATA

2, 10, 100, 1000, 3445, 3454, 3544, 4345, 4354, 4435, 4453, 4534, 4543, 5344, 5434, 5443, 10000, 12567, 12576, 12657, 12675, 12756, 12765, 15267, 15276, 15627, 15672, 15726, 15762, 16257, 16275, 16527, 16572, 16725, 16752, 17256, 17265, 17526, 17562, 17625, 17652, 21567, 21576

OFFSET

1,1

EXAMPLE

22354 is in the sequence because (2+2+3+5+4) + (2*2*3*5*4) = (2+2+3+5+4)^2

= 16^2.

MAPLE

with(numtheory):for n from 1 to 50000 do:l:=length(n):n0:=n:s:=0:p:=1:for m

from 1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v :s:=s+u:p:=p*u:od:if

s+p=s^2 then printf(`%d, `, n):else fi:od:

KEYWORD

nonn,base,less,changed

recycled

AUTHOR

Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 21 2011