A283427 a(n) is the number of consecutive smallest prime totatives of primorial A002110(n).

0, 1, 7, 26, 34, 55, 65, 91, 137, 152, 208, 251, 270, 315, 394, 471, 502, 591, 656, 685, 790, 864, 977, 1139, 1227, 1268, 1354, 1395, 1494, 1847, 1945, 2109, 2157, 2455, 2512, 2693, 2878, 3005, 3202, 3396, 3471, 3826, 3902, 4045, 4119, 4581, 5059, 5226, 5307

Offset: 1

Author

Jamie Morken and Michael De Vlieger, May 15 2017

Keywords

• nonn
• easy

Comments

Let p_n# = A002110(n) be the n-th primorial, and let t be a totative of p_n#, i.e., gcd(t, p_n#) = 1. Let q be the smallest prime totative of p_n#. We know q must be p_(n+1) by the definition of "primorial" as the product of the smallest n primes. This is the starting point of the range of primes we are considering. The ending point is the smallest composite totative, which is a square semiprime. This semiprime in fact must be q^2, since q is the smallest prime totative of p_n#. Stated in terms of prime n, the range we are considering are primes p_(n+1) <= t <= prevprime((p_(n+1))^2). For the smallest primorials, q^2 > p_n# with n <= 3. Thus a(n) < A054272(n) for n <= 3.

Examples

			a(2) = pi(min(prime(3)^2, p_2#)) - 2 = pi(min(25,6)) - 2 = 3 - 2 = 1.
a(4) = pi(min(prime(5)^2, p_4#)) - 4 = pi(min(121,210)) - 4 = 30 - 4 = 26.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000849, A054272, A283425.

Programs

Mathematica

Table[PrimePi[Min[Prime[n + 1]^2, Product[Prime@ i, {i, n}]]] - n, {n, 49}] (* Michael De Vlieger, May 16 2017 *)

Formula

a(n) = pi(min(prime(n+1)^2, Product_{k=1..n} ( prime(k) ) )) - n.