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%I #15 Jul 30 2022 08:18:43
%S 1,1,2,1,1,2,8,3,1,2,1,6,4,1,1,2,1,2,1,1,12,1,1,4,16,10,1,1,18,8,3,1,
%T 4,1,2,5,2,27,1,2,1,6,1,32,14,3,1,1,1,2,4,1,8,25,128,1,2,9,2,4,1,2,3,
%U 1,4,2,1,8,1,2,16,1,1,2,9,1,2,6,40,4,1,2,1
%N a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.
%C f(n) = a(n)/A356094(n) is the asymptotic density of numbers k such that prime(n) = A053669(k) is the smallest prime not dividing k.
%C The asymptotic mean of A053669 is 2.9200509773... (A249270) which is the weighted mean of the primes with f(n) as weights. The corresponding asymptotic standard deviation, which can be evaluated from the second raw moment Sum_{n>=1} f(n) * prime(n)^2, is 2.8013781465... .
%H Amiram Eldar, <a href="/A356093/b356093.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 1 iff prime(n) is in A039787.
%F Let f(n) = a(n)/A356094(n):
%F f(n) = A006093(n)/A002110(n).
%F Sum_{n>=1} f(n) = 1.
%F Sum_{n>=1} f(n) * prime(n) = A249270.
%e Fractions begin with 1/2, 1/3, 2/15, 1/35, 1/231, 2/5005, 8/255255, 3/1616615, 1/10140585, 2/462120945, ...
%t primorial[n_] := Product[Prime[i], {i, 1, n}]; Numerator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 100}]]
%o (PARI) a(n) = numerator((prime(n)-1)/factorback(primes(n))); \\ _Michel Marcus_, Jul 26 2022
%o (Python)
%o from math import gcd
%o from sympy import prime, primorial
%o def A356093(n): return (p:=prime(n)-1)//gcd(p,primorial(n)) # _Chai Wah Wu_, Jul 26 2022
%Y Cf. A002110, A006093, A039787, A053669, A249270, A356094 (denominators).
%Y Similar sequences: A038110, A338559, A340818, A341431, A342450, A342479.
%K nonn,frac
%O 1,3
%A _Amiram Eldar_, Jul 26 2022