A174909 -id:A174909 - cp's OEIS frontend

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

Labos Elemer, Feb 28 2000

nonn

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

			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).

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

Cf. A002110, A051953, A005867, A038110, A038111, A174909, A293558.
Cf. A000040 (prime numbers).
Cf. A161527, A060753.
Column 1 of A281891.

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 *)

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

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

A293558 Triangle T(n,k) read by rows: T(n,k) = A005867(k-1)*A002110(n)/A002110(k).

Original entry on oeis.org

1, 3, 1, 15, 5, 2, 105, 35, 14, 8, 1155, 385, 154, 88, 48, 15015, 5005, 2002, 1144, 624, 480, 255255, 85085, 34034, 19448, 10608, 8160, 5760, 4849845, 1616615, 646646, 369512, 201552, 155040, 109440, 92160

Offset: 1

Bob Selcoe, Oct 11 2017

T(n,k) is the triangle in A174909 with reversed row order. (See that sequence for additional comments).
Row sums = A053144(n) = A002110(n) - T(n+1,n+1).
T(n,k) = number of terms with smallest prime factor prime(k) contained in primorial(n) consecutive numbers, k <= n. For example, T(5,4) = 88, so there are 88 terms with smallest prime factor 7 in any sequence of 2310 consecutive numbers.

			Triangle starts:
n/k  1     2    3    4    5    6
1    1
2    3     1
3    15    5    2
4    105   35   14   8
5    1155  385  154  88   48
6    15015 5005 2002 1144 624 480
T(5,3) = 154: A005867(2) = 2, A002110(5) = 2310, A002110(3) = 30; 2*2310/30 = 154.

Cf. A000040 (prime numbers), A002110, A005867, A053144, A174909 (this triangle with reversed row order).

Table[#1 Product[EulerPhi@ Prime@ i, {i, k - 1}]/#2 & @@ Map[Product[ Prime@ i, {i, #}] &, {n, k}], {n, 8}, {k, n}] // Flatten (* Michael De Vlieger, Oct 12 2017 *)

cp's OEIS Frontend

A053144 Cototient of the n-th primorial number.

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