A259832 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) = sigma(n).

7708, 9020, 86934, 92128, 120228, 325180, 372000, 491630, 565724, 739032, 862780, 1120024, 1344090, 1419304, 1440858, 1678232, 2752626, 2980515, 3684344, 4154418, 4860476, 7539610, 7565257, 9527064, 11025372, 12277728, 17002336, 20256672, 22528536, 24597984

Offset: 1

Author

Paolo P. Lava, Jul 06 2015

Keywords

• nonn
• base

Comments

a(62) > 10^8. - Hiroaki Yamanouchi, Sep 24 2015

Examples

			7708 in base 2 is 1111000011100. If we take 1111000011100 = concat(11110000, 11100) then 11110000 and 11100 converted to base 10 are 240 and 28. Finally (sigma(240) - 240)*(sigma(28) - 28) = (744 - 240)*(56 - 28) = 504 * 28 = 14112 = sigma(7708); 9020 in base 2 is 10001100111100. If we take 10001100111100= concat(10001100, 111100) then 110 and 01111110000 converted to base 10 are 140 and 60. Finally (sigma(140) - 140)*(sigma(60) - 60) = (336 - 140)*(168 - 60)= 196 * 108 = 21160 = sigma(9020).
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..61

Crossrefs

Cf. A000203, A244079, A258813, A258843, A259831.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 1 to q do c:=convert(n, binary, decimal);
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)=sigma(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@ 125000, MemberQ[(DivisorSigma[1, #1] - #1) (DivisorSigma[1, #2] - #2) & @@@ f@ #, DivisorSigma[1, #]] &] (* Michael De Vlieger, Jul 07 2015 *)

Python

from sympy import divisor_sigma
A259832_list= []
for n in range(2,10**6):
    s, k = format(n,'0b'), divisor_sigma(n)
    for l in range(1,len(s)):
        n1, n2 = int(s[:l],2), int(s[l:],2)
        if n2 > 0 and k == (divisor_sigma(n1)-n1)*(divisor_sigma(n2)-n2):
            A259832_list.append(n)
            break # Chai Wah Wu, Jul 17 2015

Extensions

a(16)-a(21) from Chai Wah Wu, Jul 17 2015

a(22)-a(30) from Hiroaki Yamanouchi, Sep 24 2015