

A169916


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


3



0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 220, 242, 264, 286, 208, 220, 242, 264, 286, 208, 440, 462, 484, 406, 428, 440, 462, 484, 406, 428, 660, 682, 604, 626, 648, 660, 682, 604, 626, 648, 880, 802, 824, 846, 868, 880, 802, 824, 846, 868, 0, 22, 44, 66, 88, 0, 22, 44, 66, 88, 220, 242
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OFFSET

0,2


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..61.
Index entries for sequences related to carryless arithmetic


FORMULA

a(n)=a(n') if respective digits of n and n' differ by 0 or 5. In particular, a(10k+m) = a(10k+m+5) if 0 <= m <= 4.


EXAMPLE

a(16) = 16*16 = 242:
....16
....16

....72 (6*6 = 6+6 mod 10 = 2, 6*1 = 6+1 mod 10 = 7)
...27.

...242



PROG

(PARI) A169916(n)={u=vector(#n=digits(n), i, 1); n=apply(d>n+d*u, n)%10; sum(i=0, 2*#n2, sum(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: A113603 A004520 A169918 * A073909 A036211 A127353
Adjacent sequences: A169913 A169914 A169915 * A169917 A169918 A169919


KEYWORD

nonn,base


AUTHOR

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


STATUS

approved



