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

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(16) = 16*16 = 242:
....16
....16
------
....72 (6*6 = 6+6 mod 10 = 2, 6*1 = 6+1 mod 10 = 7)
...27.
------
...242
------
		

Links

• Index entries for sequences related to carryless arithmetic

Crossrefs

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

Programs

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

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.