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

1,2

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, ...

Links

Table of n, a(n) for n=1..20.

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

Adjacent sequences: A166480 A166481 A166482 * A166484 A166485 A166486

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