A169917 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be multiplication mod 10.

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 111, 144, 199, 166, 155, 166, 199, 144, 111, 400, 441, 464, 469, 446, 405, 446, 469, 464, 441, 900, 991, 964, 919, 946, 955, 946, 919, 964, 991, 600, 661, 644, 649, 666, 605, 666, 649, 644, 661, 500, 551, 504, 559, 506, 555, 506, 559, 504

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(24) = 24*24 = 446:
...24
...24
-----
...86
..48.
-----
..446
(The rule for "adding" the columns is to multiply mod 10: 8+8 = 8 * 8 mod 10 = 4.)
		

Links

• Index entries for sequences related to carryless arithmetic

Crossrefs

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

Programs

PARI

A169917(n)={#n=digits(n);n=apply(d->n*d,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

Formula

a(n) = a(n') if the i-th digit of n' either equals the i-th digit of n or (10 - the i-th digit of n): e.g., a(12345) = a(18365), because the 2nd and 4th digit of 12345 equal 10-(the 2nd resp. 4th digit of 18365), and the other digits are the same. In particular, a(10k+5+m) = a(10k+5-m), for m=0,...,4. - M. F. Hasler, Mar 26 2015