cp's OEIS Frontend

A053144 Cototient of the n-th primorial number.

%I A053144 #61 Jun 18 2022 14:27:22
%S A053144 1,4,22,162,1830,24270,418350,8040810,186597510,5447823150,
%T A053144 169904387730,6317118448410,260105476071210,11228680258518030,
%U A053144 529602053223499410,28154196550210460730,1665532558389396767070
%N A053144 Cototient of the n-th primorial number.
%C A053144 a(n) > A005367(n), a(n) > A002110(n)/2.
%C A053144 Limit_{n->oo} a(n)/A002110(n) = 1 because (in the limit) the quotient is the probability that a randomly selected integer contains at least one of the first n primes in its factorization. - _Geoffrey Critzer_, Apr 08 2010
%H A053144 Michael De Vlieger, <a href="/A053144/b053144.txt">Table of n, a(n) for n = 1..350</a>
%F A053144 a(n) = A051953(A002110(n)) = A002110(n) - A005867(n).
%F A053144 a(n) = a(n-1)*A000040(n) + A005867(n-1). - _Bob Selcoe_, Feb 21 2016
%F A053144 a(n) = (1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1). - _Jamie Morken_, Feb 08 2019
%F A053144 a(n) = A161527(n)*A002110(n)/A060753(n+1). - _Jamie Morken_, May 13 2022
%e A053144 In the reduced residue system of q(4) = 2*3*5*7 - 210 the number of coprimes to 210 is 48, while a(4) = 210 - 48 = 162 is the number of values divisible by one of the prime factors of q(4).
%t A053144 Abs[Table[ Total[Table[(-1)^(k + 1)* Total[Apply[Times, Subsets[Table[Prime[n], {n, 1, m}], {k}], 2]], {k, 0, m - 1}]], {m, 1, 22}]] (* _Geoffrey Critzer_, Apr 08 2010 *)
%t A053144 Array[# - EulerPhi@ # &@ Product[Prime@ i, {i, #}] &, 17] (* _Michael De Vlieger_, Feb 17 2019 *)
%o A053144 (PARI) a(n) = prod(k=1, n, prime(k)) - prod(k=1, n, prime(k)-1); \\ _Michel Marcus_, Feb 08 2019
%Y A053144 Cf. A002110, A051953, A005867, A038110, A038111, A174909, A293558.
%Y A053144 Cf. A000040 (prime numbers).
%Y A053144 Cf. A161527, A060753.
%Y A053144 Column 1 of A281891.
%K A053144 nonn
%O A053144 1,2
%A A053144 _Labos Elemer_, Feb 28 2000