OFFSET
1,1
COMMENTS
a(31) > 5*10^8. - Hiroaki Yamanouchi, Sep 24 2015
LINKS
Hiroaki Yamanouchi, Table of n, a(n) for n = 1..30
EXAMPLE
216 in base 2 is 11011000. If we take 11011000 = concat(110,11000) then 110 and 11000 converted to base 10 are 6 and 24. Finally (sigma(6) - 6)*(sigma(24) - 24) = (12 - 6)*(60 - 24) = 6 * 36 = 216;
13296 in base 2 is 11001111110000. If we take 11001111110000 = concat(110,01111110000) then 110 and 01111110000 converted to base 10 are 6 and 1008. Finally (sigma(6) - 6)*(sigma(1008) - 1008) = (12 - 6)*(3224 - 1008)= 6 * 2216 = 13296.
MAPLE
with(numtheory): P:=proc(q) local a, b, c, j, k, n;
for n from 1 to q do c:=convert(n, binary, decimal);
j:=0; for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k), decimal, binary);
b:=convert((c mod 10^k), decimal, binary);
if a*b>0 then if (sigma(a)-a)*(sigma(b)-b)=n then print(n);
break; fi; fi; od; od; end: P(10^9);
MATHEMATICA
f[n_] := Block[{d = IntegerDigits[n, 2], len = IntegerLength[n, 2], k}, ReplaceAll[Reap[Do[k = {FromDigits[Take[d, i], 2], FromDigits[Take[d, -(len - i)], 2]}; If[! MemberQ[k, 0], Sow@ k], {i, 1, len - 1}]], {} -> {1}][[-1, 1]]]; Select[Range@ 100000, MemberQ[(DivisorSigma[1, #1] - #1) (DivisorSigma[1, #2] - #2) & @@@ f@ #, #] &] (* Michael De Vlieger, Jul 07 2015 *)
PROG
(Python)
from sympy import divisor_sigma
A259831_list= []
for n in range(2, 10**6):
....s = format(n, '0b')
....for l in range(1, len(s)):
........n1, n2 = int(s[:l], 2), int(s[l:], 2)
........if n2 > 0 and n == (divisor_sigma(n1)-n1)*(divisor_sigma(n2)-n2):
............A259831_list.append(n)
............break # Chai Wah Wu, Jul 17 2015
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Jul 06 2015
EXTENSIONS
a(13)-a(14) from Chai Wah Wu, Jul 17 2015
a(15)-a(28) from Hiroaki Yamanouchi, Sep 24 2015
STATUS
approved