cp's OEIS Frontend

A166483 Number of integers in [1..A112141(n)] that are multiples of the first n semiprimes, where A112141(n) is the product of the first n semiprimes.

%I A166483 #18 Mar 25 2024 21:14:07
%S A166483 1,8,84,912,13344,210240,4536000,101969280,2619086400,69184886400,
%T A166483 2314236355200,79546238956800,2824874214624000,108310887381312000,
%U A166483 4266362489957568000,197608471322790528000,9780059370744254592000,502210378611975825792000,27829868062695415516800000,1595350685432248426281600000
%N A166483 Number of integers in [1..A112141(n)] that are multiples of the first n semiprimes, where A112141(n) is the product of the first n semiprimes.
%C A166483 Previous name: Numerator of sequence of ratios of semiprimes and multiples thereof in intervals of length 4*6, 4*6*9, 4*6*9*10, ... [A112141] to the length of the interval. See example below. The first few ratios are 1/4, 8/24, 84/216, 912/2160, ...
%C A166483 Conjecture: sequences for 2-ads, 3-ads (composites of 3 primes), 4-ads, etc., converge to 1/2, 1/4, 1/8, ..., respectively.
%C A166483 The first few analogous ratios for 3-ads are 1/8, 16/96, 336/1728, 7296/34560, ...
%e A166483 For n=2, the product of first two semiprimes is 4*6 = 24, and there are 8 multiples of the semiprimes 4,6 in the interval [1..24] including those two semiprimes, with repetitions (such as 4*3, 2*6) counted only once. So a(2) = 8.
%p A166483 f:= proc(n)
%p A166483   local N, V, T, S;
%p A166483   N:= convert(SP[1..n],`*`);
%p A166483   T:= 0:
%p A166483   for S in combinat:-powerset(convert(SP[1..n],set)) minus {{}} do
%p A166483     if nops(S)::odd then T:= T + N/ilcm(op(S)) else T:= T - N/ilcm(op(S)) fi
%p A166483   od;
%p A166483   T
%p A166483 end proc:
%p A166483 map(f, [$1..20]); # _Robert Israel_, Mar 24 2024
%o A166483 (PARI) ismul(x, v) = for (k=1, #v, if ((x % v[k]) == 0, return(1)););
%o A166483 vsp(n) = my(v=vector(n), i, k=3); while(i<n, if(bigomega(k++)==2, v[i++]=k)); v;
%o A166483 a(n) = my(v = vsp(n)); sum(k=1, vecprod(v), ismul(k, v)); \\ _Michel Marcus_, Aug 16 2022
%Y A166483 Cf. A112141 (product of the first n semiprimes).
%K A166483 nonn
%O A166483 1,2
%A A166483 _Daniel Tisdale_, Oct 14 2009
%E A166483 a(2) and a(6) corrected, and new name from _Michel Marcus_, Aug 15 2022
%E A166483 More terms from _Robert Israel_, Mar 24 2024