A307826 The number of integers r such that all primes above a certain value have the form primorial(n)*q +- r.

1, 1, 4, 24, 240, 2880, 46080, 829440, 18247680, 510935040, 15328051200, 551809843200, 22072393728000, 927040536576000, 42643864682496000, 2217480963489792000, 128613895882407936000, 7716833752944476160000, 509311027694335426560000

Offset: 1

Author

William Boyles, Apr 30 2019

Keywords

• nonn

Examples

			For n=3, the third primorial is 2*3*5=30, and all primes at least 17 have the form 30n +- (1,7,11,13). So, a(3) = 4.
		

Links

• William Boyles, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A000010, A002110, A005867, A156037.

Programs

Mathematica

a[1]=1; a[n_] := EulerPhi[Product[Prime[i], {i, 1, n}]]/2; Array[a, 20] (* Amiram Eldar, Jul 08 2019 *)

Python

import sympy
def A307826(n):
    sympy.sieve.extend_to_no(n)
    s = list(sympy.sieve._list)
    prod = s[0]
    print("1")
    for i in range(1,n):
        prod*=s[i]
        print(sympy.ntheory.factor_.totient(prod)//2)

Formula

a(n) = Product_{k=1..n} A156037(k).

a(n) = A000010(A002110(n))/2 for n > 1.

a(n) = A005867(n)/2 for n > 1. - Alexandre Herrera, Apr 16 2023