Octave, 62 61 bytes

@(n)sum(arrayfun(@(t)any(strfind((g=@dec2bin)(n),g(t))),1:n))

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###Explanation

For input n, the code tests all numbers from 1 to n to see if their binary representation is a substring of the binary representation of the input.

@(n)                                                          % Anonymous function of n
        arrayfun(                                      ,1:n)  % Map over range 1:n
                 @(t)                                         % Anonymous function of t
                         strfind(               ,    )        % Indices of ...
                                                 g(t)         % t as binary string ...
                                 (g=@dec2bin)(n)              % within n as binary string
                     any(                             )       % True if contains nonzero
    sum(                                                    ) % Sum of array

Octave, 62 61 bytes

@(n)sum(arrayfun(@(t)any(strfind((g=@dec2bin)(n),g(t))),1:n))

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Explanation

For input n, the code tests all numbers from 1 to n to see if their binary representation is a substring of the binary representation of the input.

@(n)                                                          % Anonymous function of n
        arrayfun(                                      ,1:n)  % Map over range 1:n
                 @(t)                                         % Anonymous function of t
                         strfind(               ,    )        % Indices of ...
                                                 g(t)         % t as binary string ...
                                 (g=@dec2bin)(n)              % within n as binary string
                     any(                             )       % True if contains nonzero
    sum(                                                    ) % Sum of array

Octave, 62 bytes

@(n)sum(arrayfun(@(t)any(strfind(dec2bin(n),dec2bin(t))),1:n))

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###Explanation

For input n, the code tests all numbers from 1 to n to see if their binary representation is a substring of the binary representation of the input.

@(n)                                                           % Anonymous function of n
        arrayfun(                                       ,1:n)  % Map over range 1:n
                 @(t)                                          % Anonymous function of t
                         strfind(          ,           )       % Indices of ...
                                            dec2bin(t)         % t as binary string ...
                                 dec2bin(n)                    % within n as binary string
                     any(                              )       % True if contains nonzero
    sum(                                                     ) % Sum of array

Octave, 62 61 bytes

@(n)sum(arrayfun(@(t)any(strfind((g=@dec2bin)(n),g(t))),1:n))

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###Explanation

For input n, the code tests all numbers from 1 to n to see if their binary representation is a substring of the binary representation of the input.

@(n)                                                          % Anonymous function of n
        arrayfun(                                      ,1:n)  % Map over range 1:n
                 @(t)                                         % Anonymous function of t
                         strfind(               ,    )        % Indices of ...
                                                 g(t)         % t as binary string ...
                                 (g=@dec2bin)(n)              % within n as binary string
                     any(                             )       % True if contains nonzero
    sum(                                                    ) % Sum of array

Octave, 62 bytes

@(n)sum(arrayfun(@(t)any(strfind(dec2bin(n),dec2bin(t))),1:n))

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###Explanation

For input n, the code tests all numbers from 1 to n to see if their binary representation is a substring of the binary representation of the input.

@(n)                                                           % Anonymous function of n
        arrayfun(                                       ,1:n)  % Map over range 1:n
                 @(t)                                          % Anonymous function of t
                         strfind(          ,           )       % Indices of ...
                                 dec2bin(n)                    % n as binary string ...
                                            dec2bin(t)         % within t as binary string
                     any(                              )       % True if contains nonzero
    sum(                                                     ) % Sum of array

Octave, 62 bytes

@(n)sum(arrayfun(@(t)any(strfind(dec2bin(n),dec2bin(t))),1:n))

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###Explanation

For input n, the code tests all numbers from 1 to n to see if their binary representation is a substring of the binary representation of the input.

@(n)                                                           % Anonymous function of n
        arrayfun(                                       ,1:n)  % Map over range 1:n
                 @(t)                                          % Anonymous function of t
                         strfind(          ,           )       % Indices of ...
                                            dec2bin(t)         % t as binary string ...
                                 dec2bin(n)                    % within n as binary string
                     any(                              )       % True if contains nonzero
    sum(                                                     ) % Sum of array