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 A161170 Least integer k such that the n-almost prime count is equal to the prime count. 0
 10, 125, 1809, 37820, 2722768, 1037849736, 4204496515890 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified August 13 06:22 EDT 2020. Contains 336442 sequences. (Running on oeis4.)