A357604 Number of prime powers in the sequence of the floor of n/k for k <= n, A010766.

0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 4, 5, 6, 6, 8, 8, 7, 8, 7, 8, 10, 11, 9, 10, 11, 12, 12, 13, 11, 12, 14, 14, 15, 16, 14, 15, 16, 17, 16, 17, 16, 17, 18, 18, 20, 21, 19, 21, 21, 21, 22, 23, 22, 23, 23, 24, 26, 27, 22, 23, 24, 25, 28, 28, 28, 29, 29, 30, 30, 31, 27

Offset: 1

Author

Randell Heyman, Oct 06 2022

Keywords

• nonn

Comments

Conjecture: a(n+1) - a(n) has all its record values at n = A135972(k) and the record values of a(n-1) - a(n) are a subsequence of A025487. This was verified for n = 1..20000. - Thomas Scheuerle, Oct 06 2022

Examples

			For n=8 we have floor(8/1) = 8 = 2^3, a prime power; floor(8/2) = 4 = 2^2, a prime power; floor(8/3) = floor(8/4) = 2 = 2^1, a prime power. Each remaining term of the sequence is 1, which is not a prime power, so a(8) = 4.
		

Links

• R. Heyman, Primes in floor function sets, Integers 22(2022), #A59.

Crossrefs

Cf. A000961, A025487, A135972, A010766.

Programs

MATLAB

function a = A357604( max_n )
    for n = 1:max_n
        s = floor(n./[1:n]); c = 0;
        for m = 1:n-1
            f = factor(s(m));
            if s(m) > 1 && length(unique(f)) == 1
                c = c+1;
            end
        end
        a(n) = c;
    end
end % Thomas Scheuerle, Oct 06 2022

PARI

a(n) = sum(k=1,n, isprimepower(n\k)!=0); \\ Thomas Scheuerle, Oct 07 2022

Formula

a(n) = c*n + O(n^(1/2)), where c is the sum of 1/(q*(q+1)) over all prime powers q.

Extensions

a(12)-a(72) from Thomas Scheuerle, Oct 06 2022