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

0,2

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

Example

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)

Prog

(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

Crossrefs

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

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

Sequence in context: A169933 A113603 A004520 * A169916 A073909 A036211

Adjacent sequences: A169915 A169916 A169917 * A169919 A169920 A169921

Keyword

nonn,base

Author

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

Extensions

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

Status

approved