A244313 - OEIS

%I #8 Jun 26 2014 18:02:02

%S 38,58,66,87,110,205,210,310,410,510,610,710,714,810,910,1010,2010,

%T 3010,4010,5010,6010,7010,8010,9010,10010,20010,30010,40010,50010,

%U 60010,70010,80010,90010,100010,200010,300010,400010,500010,600010,700010,800010,900010

%N Consider a number n with m decimal digits, the prefix p of length m-1 and the suffix s of length m-1. The sequence lists the numbers n such that sigma(n) = sigma(p)*sigma(s) where sigma(n) is the sum of the divisors of n.

%C Property of the sequence :

%C {a(n)} = E1 union E2 where E1 = {38, 58, 66, 87, 205, 714} and E2 = {110, 210, 310, 410, 510, 610, 710, 810, 910, 1010, 2010,...}.

%C E2 = F1 union F2 union... union Fk union... where :

%C F1 = {110, 210,..., 910} with one zero;

%C F2 = {1010, 2010,..., 9010} with two zeros;

%C F3 = {10010, 20010,..., 90010} with three zeros;

%C ......................................................................

%C Fk = {100…0010, 200…0010,..., 900...0010} with k zeros;

%C ......................................................................

%C Hence the proposition :

%C If n is of the form n = a0000...010 with k zeros and a =1,2,..., 9 then sigma(n) = sigma(x)*sigma(y) where x = a0000...01 with k-1 zeros and y = 10.

%e 3010 is in the sequence because sigma(3010) = 6336; sigma(301) = 352 and sigma(10) = 18 => 6336 = 352*18.

%p with(numtheory):

%p for n from 10 to 10000 do:

%p       x:=convert(n, base, 10):n1:=nops(x):

%p       s1:=sum('x[i]*10^(i-1) ', 'i'=1..n1-1):

%p       s2:=(n-irem(n,10))/10:

%p       x1:=sigma(s1):x2:=sigma(s2):xn:=sigma(n):

%p       if xn = x1*x2

%p         then

%p         printf(`%d, `, n):

%p         else

%p       fi:

%p od:

%Y Cf. A000203.

%K nonn,base

%O 1,1

%A _Michel Lagneau_, Jun 25 2014