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.

1, 8, 84, 912, 13344, 210240, 4536000, 101969280, 2619086400, 69184886400, 2314236355200, 79546238956800, 2824874214624000, 108310887381312000, 4266362489957568000, 197608471322790528000, 9780059370744254592000, 502210378611975825792000, 27829868062695415516800000, 1595350685432248426281600000

Offset: 1

Author

Daniel Tisdale, Oct 14 2009

Keywords

• nonn

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, ...

Examples

			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.
		

Crossrefs

Cf. A112141 (product of the first n semiprimes).

Programs

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

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(iMichel Marcus, Aug 16 2022

Extensions

a(2) and a(6) corrected, and new name from Michel Marcus, Aug 15 2022

More terms from Robert Israel, Mar 24 2024