A169918 Squares in carryless arithmetic mod 10 with addition of digits defined to be multiplication mod 10 and multiplication of digits defined to be addition mod 10.

0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 210, 242, 294, 266, 258, 260, 292, 244, 216, 208, 440, 492, 464, 456, 468, 490, 442, 414, 406, 418, 690, 662, 654, 666, 698, 640, 612, 604, 616, 648, 860, 852, 864, 896, 848, 810, 802, 814, 846, 898, 50, 62, 94, 46, 18, 0, 12, 44, 96, 68, 260, 292

Offset: 0

Author

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

Keywords

• nonn
• base

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.

Examples

			a(17) = 17*17 = 244:
...17
...17
-----
...84 (7*7 = 7+7 mod 10 = 4, 7*1 = 7+1 mod 10 = 8)
..28.
-----
..244
(The rule for "adding" the columns is to multiply mod 10: 8+8 = 8 * 8 mod 10 = 4. Blanks are ignored)
		

Links

• Index entries for sequences related to carryless arithmetic

Crossrefs

See A048379, A169931-A169933, A169935 for other examples of calculations in this version of arithmetic.

The four versions are A059729, A169916, A169917, A169918.

Programs

PARI

A169918(n)={u=vector(#n=digits(n),i,1);n=apply(d->n+d*u,n)%10;sum(i=0,2*#n-2,prod(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

Extensions

Thanks to Rick L. Shepherd for pointing out a typo in the example. - N. J. A. Sloane, Nov 08 2014