A169908 a(n) = n*n in carryless digital root arithmetic in base 10.

0, 1, 4, 9, 7, 7, 9, 4, 1, 9, 100, 121, 144, 169, 187, 117, 139, 154, 171, 199, 400, 441, 484, 439, 477, 427, 469, 414, 451, 499, 900, 961, 934, 999, 967, 937, 999, 964, 931, 999, 700, 781, 774, 769, 757, 747, 739, 724, 711, 799, 700, 711, 724, 739, 747, 757

Offset: 0

Author

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 11 2010

Keywords

• nonn
• base

Comments

Addition and multiplication are the same as in school, that is, done in base 10, except that there are no carries and when individual digits are added or multiplied the result is replaced by its digital root (A010888).

Examples

			14*14 = 187:
..14
..14
----
..47
.14.
----
.187
----
		

Links

• Index entries for sequences related to carryless arithmetic

Crossrefs

Cf. A010888, A169821.

Programs

Maple

A010888 := proc(n)
    if n = 0 then
        0;
    else
        1+modp(n-1,9) ;
    end if;
end proc:
carryLmult1dig := proc(a,b)
    local adigs,cdigs,d,len ;
    if b <= 9 then
        adigs := convert(a,base,10) ;
        len := nops(adigs) ;
        cdigs := [seq( A010888(b*op(d,adigs)) ,d=1..len)] ;
        add(op(d,cdigs)*10^(d-1),d=1..len) ;
    else
        error b,"not smaller than 10" ;
    end if;
end proc:
carryLmult := proc(a,b)
    local adigs,bdigs,c,len ;
    adigs := convert(a,base,10) ;
    bdigs := convert(b,base,10) ;
    len := max(nops(adigs),nops(bdigs)) ;
    c := 0 ;
    for i from 1 to nops(bdigs) do
        carryLmult1dig(a,op(i,bdigs)) ;
        c := carryLadd(c,%*10^(i-1)) ;
    end do:
    c ;
end proc:
A169908 := proc(n)
    carryLmult(n,n) ;
end proc: # R. J. Mathar, Jul 12 2013