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

0,3

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..58.

Index entries for sequences related to carryless arithmetic

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

Example

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.)

Prog

(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

Crossrefs

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

Sequence in context: A186723 A008959 A316347 * A059729 A184988 A108533

Adjacent sequences: A169914 A169915 A169916 * A169918 A169919 A169920

Keyword

nonn,base

Author

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

Status

approved