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Carryless sum of divisors of n.
3

%I #16 Jan 18 2019 13:00:17

%S 1,3,4,7,6,2,8,5,3,18,12,18,14,14,14,11,18,19,10,32,22,36,24,30,21,32,

%T 20,36,20,52,32,43,48,44,38,51,38,40,46,70,42,76,44,74,58,62,48,84,47,

%U 83,62,88,54,80,62,80,60,70,50,48,62,96,84,7,74,24,68,6

%N Carryless sum of divisors of n.

%C This sequence is a variant of A178910 for the base 10.

%H Rémy Sigrist, <a href="/A323394/b323394.txt">Table of n, a(n) for n = 1..19999</a>

%H Rémy Sigrist, <a href="/A323394/a323394.png">Scatterplot of the first 1000000 terms</a>

%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) <= A000203(n).

%e For n = 42:

%e - the divisors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42,

%e - the sum of the units is: 1 + 2 + 3 + 6 + 7 + 4 + 1 + 2 = 26 == 6 (mod 10),

%e - the sum of the tens is: 1 + 2 + 4 = 7,

%e - hence a(42) = 76.

%e For n = 973:

%e - the divisors of 973 are: 1, 7, 139, 973,

%e - the sum of the units is: 1 + 7 + 9 + 3 = 20 == 0 (mod 10),

%e - the sum of the tens is: 3 + 7 = 10 == 0 (mod 10),

%e - the sum of the hundreds is: 1 + 9 = 10 == 0 (mod 10),

%e - hence a(973) = 0.

%p f:= proc(n) local t,d,dd,m,i;

%p t:= Vector(convert(n,base,10));

%p for d in numtheory:-divisors(n) minus {n} do

%p dd:= convert(d,base,10);

%p m:= nops(dd);

%p t[1..m]:= t[1..m] + Vector(dd) mod 10;

%p od:

%p add(t[i]*10^(i-1),i=1..ilog10(n)+1)

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Jan 15 2019

%o (PARI) a(n, base=10) = my (v=[]); fordiv (n, d, my (w=Vecrev(digits(d, base))); v=vector(max(#v, #w), k, (if (k>#v, w[k], k>#w, v[k], (v[k]+w[k])%base)))); fromdigits(Vecrev(v), base)

%Y Cf. A000203, A169890, A178910, A323414 (positions of zeros), A323415 (fixed points).

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Jan 13 2019