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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.
0
1, 8, 84, 912, 13344, 210240, 4536000, 101969280, 2619086400, 69184886400, 2314236355200, 79546238956800, 2824874214624000, 108310887381312000, 4266362489957568000, 197608471322790528000, 9780059370744254592000, 502210378611975825792000, 27829868062695415516800000, 1595350685432248426281600000
OFFSET
1,2
COMMENTS
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, ...
Conjecture: sequences for 2-ads, 3-ads (composites of 3 primes), 4-ads, etc., converge to 1/2, 1/4, 1/8, ..., respectively.
The first few analogous ratios for 3-ads are 1/8, 16/96, 336/1728, 7296/34560, ...
EXAMPLE
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.
MAPLE
f:= proc(n)
local N, V, T, S;
N:= convert(SP[1..n], `*`);
T:= 0:
for S in combinat:-powerset(convert(SP[1..n], set)) minus {{}} do
if nops(S)::odd then T:= T + N/ilcm(op(S)) else T:= T - N/ilcm(op(S)) fi
od;
T
end proc:
map(f, [$1..20]); # Robert Israel, Mar 24 2024
PROG
(PARI) ismul(x, v) = for (k=1, #v, if ((x % v[k]) == 0, return(1)); );
vsp(n) = my(v=vector(n), i, k=3); while(i<n, if(bigomega(k++)==2, v[i++]=k)); v;
a(n) = my(v = vsp(n)); sum(k=1, vecprod(v), ismul(k, v)); \\ Michel Marcus, Aug 16 2022
CROSSREFS
Cf. A112141 (product of the first n semiprimes).
Sequence in context: A264710 A143868 A130591 * A048665 A005797 A233835
KEYWORD
nonn
AUTHOR
Daniel Tisdale, Oct 14 2009
EXTENSIONS
a(2) and a(6) corrected, and new name from Michel Marcus, Aug 15 2022
More terms from Robert Israel, Mar 24 2024
STATUS
approved