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

%I A307826 #46 May 15 2023 08:42:58
%S A307826 1,1,4,24,240,2880,46080,829440,18247680,510935040,15328051200,
%T A307826 551809843200,22072393728000,927040536576000,42643864682496000,
%U A307826 2217480963489792000,128613895882407936000,7716833752944476160000,509311027694335426560000
%N A307826 The number of integers r such that all primes above a certain value have the form primorial(n)*q +- r.
%H A307826 William Boyles, <a href="/A307826/b307826.txt">Table of n, a(n) for n = 1..350</a>
%F A307826 a(n) = Product_{k=1..n} A156037(k).
%F A307826 a(n) = A000010(A002110(n))/2 for n > 1.
%F A307826 a(n) = A005867(n)/2 for n > 1. - _Alexandre Herrera_, Apr 16 2023
%e A307826 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.
%t A307826 a[1]=1; a[n_] := EulerPhi[Product[Prime[i], {i, 1, n}]]/2; Array[a, 20] (* _Amiram Eldar_, Jul 08 2019 *)
%o A307826 (Python)
%o A307826 import sympy
%o A307826 def A307826(n):
%o A307826     sympy.sieve.extend_to_no(n)
%o A307826     s = list(sympy.sieve._list)
%o A307826     prod = s[0]
%o A307826     print("1")
%o A307826     for i in range(1,n):
%o A307826         prod*=s[i]
%o A307826         print(sympy.ntheory.factor_.totient(prod)//2)
%Y A307826 Cf. A000010, A002110, A005867, A156037.
%K A307826 nonn
%O A307826 1,3
%A A307826 _William Boyles_, Apr 30 2019