A169916 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be addition mod 10.

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

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*#n-2, sum(j=max(1, #n-i), min(2*#n-1-i, #n), n[2*#n-i-j][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