A259831 Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have (sigma(a)-a)*(sigma(b)-b) = n.

216, 13296, 13464, 14416, 51480, 235200, 575484, 578592, 585000, 1032656, 1121400, 1599552, 4190364, 4786110, 8365968, 11268688, 13010634, 13253436, 21835624, 22108784, 23896320, 136311840, 152820243, 160380496, 170073324, 295999900, 421686580, 445421664

Offset: 1

Author

Paolo P. Lava, Jul 06 2015

Keywords

• nonn
• base

Comments

a(31) > 5*10^8. - Hiroaki Yamanouchi, Sep 24 2015

Examples

			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.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..30

Crossrefs

Cf. A001065, A244079, A258813, A258843, A259832.

Programs

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 *)

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

Extensions

a(13)-a(14) from Chai Wah Wu, Jul 17 2015

a(15)-a(28) from Hiroaki Yamanouchi, Sep 24 2015