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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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;

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

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

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Last modified September 20 17:16 EDT 2020. Contains 337265 sequences. (Running on oeis4.)