

A357604


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


1



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
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OFFSET

1,4


COMMENTS

Conjecture: a(n+1)  a(n) has all its record values at n = A135972(k) and the record values of a(n1)  a(n) are a subsequence of A025487. This was verified for n = 1..20000.  Thomas Scheuerle, Oct 06 2022


LINKS



FORMULA

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


EXAMPLE

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.


PROG

(MATLAB)
for n = 1:max_n
s = floor(n./[1:n]); c = 0;
for m = 1:n1
f = factor(s(m));
if s(m) > 1 && length(unique(f)) == 1
c = c+1;
end
end
a(n) = c;
end
(PARI) a(n) = sum(k=1, n, isprimepower(n\k)!=0); \\ Thomas Scheuerle, Oct 07 2022


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



