A053144 Cototient of the n-th primorial number.

1, 4, 22, 162, 1830, 24270, 418350, 8040810, 186597510, 5447823150, 169904387730, 6317118448410, 260105476071210, 11228680258518030, 529602053223499410, 28154196550210460730, 1665532558389396767070

Offset

1,2

Comments

a(n) > A005367(n), a(n) > A002110(n)/2.

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..350

Formula

a(n) = A051953(A002110(n)) = A002110(n) - A005867(n).

a(n) = a(n-1)*A000040(n) + A005867(n-1). - Bob Selcoe, Feb 21 2016

a(n) = (1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1). - Jamie Morken, Feb 08 2019

a(n) = A161527(n)*A002110(n)/A060753(n+1). - Jamie Morken, May 13 2022

Example

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).

Mathematica

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 *)
Array[# - EulerPhi@ # &@ Product[Prime@ i, {i, #}] &, 17] (* Michael De Vlieger, Feb 17 2019 *)

Prog

(PARI) a(n) = prod(k=1, n, prime(k)) - prod(k=1, n, prime(k)-1); \\ Michel Marcus, Feb 08 2019

Crossrefs

Cf. A002110, A051953, A005867, A038110, A038111, A174909, A293558.

Cf. A000040 (prime numbers).

Cf. A161527, A060753.

Column 1 of A281891.

Sequence in context: A346968 A184942 A000779 * A089464 A111343 A302908

Adjacent sequences: A053141 A053142 A053143 * A053145 A053146 A053147

Keyword

nonn

Author

Labos Elemer, Feb 28 2000

Status

approved