

A169917


Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be multiplication mod 10.


3



0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 111, 144, 199, 166, 155, 166, 199, 144, 111, 400, 441, 464, 469, 446, 405, 446, 469, 464, 441, 900, 991, 964, 919, 946, 955, 946, 919, 964, 991, 600, 661, 644, 649, 666, 605, 666, 649, 644, 661, 500, 551, 504, 559, 506, 555, 506, 559, 504
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OFFSET

0,3


COMMENTS

The rules of arithmetic used in A169916, A169917, A169918 have very strange consequences. Many of the familiar laws fail. For instance, the arithmetic in A169916 is not associative: 10*(9*2) = 10*1 = 21 != (10*9)*2 = 9*2 = 1.


LINKS

Table of n, a(n) for n=0..58.
Index entries for sequences related to carryless arithmetic


FORMULA

a(n) = a(n') if the ith digit of n' either equals the ith digit of n or (10  the ith digit of n): e.g., a(12345) = a(18365), because the 2nd and 4th digit of 12345 equal 10(the 2nd resp. 4th digit of 18365), and the other digits are the same. In particular, a(10k+5+m) = a(10k+5m), for m=0,...,4.  M. F. Hasler, Mar 26 2015


EXAMPLE

a(24) = 24*24 = 446:
...24
...24

...86
..48.

..446
(The rule for "adding" the columns is to multiply mod 10: 8+8 = 8 * 8 mod 10 = 4.)


PROG

(PARI) A169917(n)={#n=digits(n); n=apply(d>n*d, n)%10; sum(i=0, 2*#n2, prod(j=max(1, #ni), min(2*#n1i, #n), n[2*#nij][j])%10*10^i)} \\ M. F. Hasler, Mar 26 2015


CROSSREFS

The four versions are A059729, A169916, A169917, A169918.
Sequence in context: A186723 A008959 A316347 * A059729 A184988 A108533
Adjacent sequences: A169914 A169915 A169916 * A169918 A169919 A169920


KEYWORD

nonn,base


AUTHOR

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010


STATUS

approved



