[seqfan] Re: Observation on the divisors of 136

[Neil Fernandez]

Hi Jeremy and seqfans,

When the divisors d(i) are in increasing order, we need
d(i+1)=2*d(i) + (0 or 1) for d(i)<i.

Mathematica code:

For[i = 1, i <= 10^7, i++,
 x = 0;
 t = Divisors[i];
 For[j = 1, j < Length[t], j++,
  If[t[[j + 1]] != 2*t[[j]] && t[[j + 1]] != 2*t[[j]] + 1, x += 1]];
 If[x == 0 && IntegerQ[Log[2, i]] == False, Print[i]]

Excluding powers of 2, the sequence begins

3, 10, 136, 32896

with no more members less than 10^7.

Observation:
3= 2^0 * (2^1 +1) = 2^0 + 2^1
10= 2^1 * (2^2 + 1) = 2^1 + 2^3
136 = 2^3 * (2^4 + 1) = 2^3 + 2^7
32896 = 2^7 * (2^8 + 1) = 2^7 + 2^15

which suggests the conjecture that all integers of the form 2^(2^k-1) +
2^(2^(k+1)-1) are in the sequence.

This conjecture is false.

2147516416 = 2^15 * (2^30 + 1) = 2^15 + 2^31 is in the sequence,
but 9223372039002259456 = 2^15 * (2^16 + 1) = 2^15 + 2^31 is not.

Neil

In message <D66C7339.2C11F%jeremy.gardiner at btinternet.com>, Jeremy
Gardiner <jeremy.gardiner at btinternet.com> writes

>Consider the divisors of 136 (A018299):
>
>1, 2, 4, 8, 17, 34, 68, 136.
>
>136 in binary is 10001000.
>
>Reading off successive bits from the left we have:
>
>1          1
>10         2
>100        4
>1000       8
>10001     17
>100010    34
>1000100   68
>10001000 136
>
>This prompts the question, what other numbers have this property?
>
>i.e. Numbers whose successive binary shifted parts as decimal numbers are
>identical to their ordered divisors.
>
>Clearly, the powers of 2 have this property.
>
>Also for example 10 in binary is 1010 giving 1, 2, 5, 10.
>
>The sequence begins:
>
>1, 2, 3, 4, 8, 10, 16, 32, ... but it is not A295296.
>
>Regards,
>
>Jeremy Gardiner
>
>
>
>--
>Seqfan Mailing list - http://list.seqfan.eu/

-- 
Neil Fernandez