A244313 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.

38, 58, 66, 87, 110, 205, 210, 310, 410, 510, 610, 710, 714, 810, 910, 1010, 2010, 3010, 4010, 5010, 6010, 7010, 8010, 9010, 10010, 20010, 30010, 40010, 50010, 60010, 70010, 80010, 90010, 100010, 200010, 300010, 400010, 500010, 600010, 700010, 800010, 900010

Offset

1,1

Comments

Property of the sequence :

{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,...}.

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

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

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

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

......................................................................

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

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Hence the proposition :

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.

Links

Table of n, a(n) for n=1..42.

Example

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

Maple

with(numtheory):
for n from 10 to 10000 do:
      x:=convert(n, base, 10):n1:=nops(x):
      s1:=sum('x[i]*10^(i-1) ', 'i'=1..n1-1):
      s2:=(n-irem(n, 10))/10:
      x1:=sigma(s1):x2:=sigma(s2):xn:=sigma(n):
      if xn = x1*x2
        then
        printf(`%d, `, n):
        else
      fi:
od:

Crossrefs

Cf. A000203.

Sequence in context: A078544 A116243 A098126 * A147616 A193568 A039466

Adjacent sequences: A244310 A244311 A244312 * A244314 A244315 A244316

Keyword

nonn,base

Author

Michel Lagneau, Jun 25 2014

Status

approved