A293558 -id:A293558 - 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

A174909 Triangle T(n,i) whose n-th row gives the number of numbers in any prime(n)# consecutive numbers whose smallest prime factor is prime(n-i+1).

Original entry on oeis.org

1, 1, 3, 2, 5, 15, 8, 14, 35, 105, 48, 88, 154, 385, 1155, 480, 624, 1144, 2002, 5005, 15015, 5760, 8160, 10608, 19448, 34034, 85085, 255255, 92160, 109440, 155040, 201552, 369512, 646646, 1616615, 4849845, 1658880, 2119680, 2517120, 3565920

Offset: 1

T. D. Noe, Apr 01 2010

Here prime(n)# denotes the product of the first n primes. Row n begins with A005867(n-1). The other n-1 terms in row n are prime(n) times the previous row. The sum of the terms in row n is cototient(prime(n)#), which is A053144(n), and which equals prime(n)#-A005867(n). This sequence is a generalization of a comment in A005867 by Dennis Martin.

			For n=3, we have prime(n)=5 and any range of 2*3*5=30 consecutive numbers has 2 numbers whose smallest prime factor is 5, 5 numbers whose smallest prime factor is 3, and 15 numbers whose smallest prime factor is 2.
From _Bob Selcoe_, Oct 12 2017: (Start)
Triangle starts:
n/i  1   2   3    4    5     6
1    1
2    1   3
3    2   5   15
4    8  14   35  105
5   48  88  154  385  1155
6  480 624 1144 2002  5005 15015
(End)

Cf. A002110, A005867 (first column), A020639, A053144, A070826 (main diagonal).

Cf. A293558 (transpose).

t={{1}}; q=2; Do[p=Prime[n]; t=AppendTo[t, Join[{(q-1)*t[[ -1,1]]}, p*t[[ -1]]]]; q=p, {n,2,9}]; Flatten[t]
(* Second program: *)
Block[{nn = 8, s}, s = Array[FactorInteger[#][[1, 1]] &, Product[Prime@i, {i, nn}]]; Table[With[{P = Product[Prime@ k, {k, n}]}, Count[Take[s, P], ?(# == Prime[n - i + 1] &)]], {n, nn}, {i, n}]] (* _Michael De Vlieger, Oct 14 2017 *)

cp's OEIS Frontend

A053144 Cototient of the n-th primorial number.

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