A161170 Least integer k such that the n-almost prime count is equal to the prime count.

10, 125, 1809, 37820, 2722768, 1037849736, 4204496515890

Offset

2,1

Comments

Related to sequence A125149, but we compare the prime count to the semiprime count, then the product-of-three-primes count, and so on.

We start with a the number two, and a prime count of 1.

Then the prime count and semiprime count are first identical when k = 10, the prime count is 4 ({2, 3, 5, 7}) and the semiprime count is also 4 ({4, 6, 9, 10}).

Next, when k = 125 the prime count of 30 and product-of-three-primes count of 30 are first identical.

Based on the write up for A125149 and examination of the factor counts as k increases, we believe this sequence is infinite, but have not proved this.

Links

Table of n, a(n) for n=2..8.

Example

a(2) = 10 since there are now 4 primes ({2, 3, 5, 7}) and 4 semiprimes ({4, 6, 9, 10}) <= 10.
a(3) = 125 with 30 primes and 30 products of 3 primes.
a(4) = 1809 with 279 primes and 279 products of 4 primes.
a(5) = 37820 with 4000 primes and 4000 products of 5 primes.
a(6) = 2722768 with 198183 primes and 198183 products of 6 primes.
a(7) = 1037849736 with 52672391 primes and 52672391 products of 7 primes.
a(8) = 4204496515890 with 150007470826 primes and 150007470826 products of 8 primes.

Prog

(Ruby) # A slow program to generate sequence
# Faster C code is available by request
# Tallies number of primes, semiprimes, trieneprimes ...
tally = Hash.new { |h, k| h[k] = 0}
# Returns number of factors of num
def factors(num)
(2..(Math.sqrt(num).to_i)).each{ |i|
return factors(num/i) + 1 if num % i == 0
}
1
end
# Testing number of primes against composites with num_factors
num_factors = 2
2.upto( 1.0/0.0 ) { |i|
tally[factors(i)] +=1
if tally[1] == tally[num_factors]
puts "k: #{i} Primes: #{tally[1]} Composites with #{num_factors} factors: #{tally[num_factors]}"
num_factors += 1
end
}
(Perl) use ntheory ":all"; my($k, @S)=(0, map{0}1..20); forfactored { $S[@_]++; while ($S[1] == $S[$k]) { print "$k $_\n" if $k>1; $k++; } } 3e6; # Dana Jacobsen, Jan 18 2019

Crossrefs

Cf. A125149.

Sequence in context: A123358 A230390 A089832 * A281595 A097816 A323877

Adjacent sequences: A161167 A161168 A161169 * A161171 A161172 A161173

Keyword

hard,more,nonn

Author

Andy Martin, Jun 04 2009

Extensions

Edited example and a(8) from Donovan Johnson, Mar 10 2010

Status

approved