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A356093 a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

%I A356093 #15 Jul 30 2022 08:18:43
%S A356093 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 A356093 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 A356093 1,4,2,1,8,1,2,16,1,1,2,9,1,2,6,40,4,1,2,1
%N A356093 a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.
%C A356093 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 A356093 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 A356093 Amiram Eldar, <a href="/A356093/b356093.txt">Table of n, a(n) for n = 1..10000</a>
%F A356093 a(n) = 1 iff prime(n) is in A039787.
%F A356093 Let f(n) = a(n)/A356094(n):
%F A356093 f(n) = A006093(n)/A002110(n).
%F A356093 Sum_{n>=1} f(n) = 1.
%F A356093 Sum_{n>=1} f(n) * prime(n) = A249270.
%e A356093 Fractions begin with 1/2, 1/3, 2/15, 1/35, 1/231, 2/5005, 8/255255, 3/1616615, 1/10140585, 2/462120945, ...
%t A356093 primorial[n_] := Product[Prime[i], {i, 1, n}]; Numerator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 100}]]
%o A356093 (PARI) a(n) = numerator((prime(n)-1)/factorback(primes(n))); \\ _Michel Marcus_, Jul 26 2022
%o A356093 (Python)
%o A356093 from math import gcd
%o A356093 from sympy import prime, primorial
%o A356093 def A356093(n): return (p:=prime(n)-1)//gcd(p,primorial(n)) # _Chai Wah Wu_, Jul 26 2022
%Y A356093 Cf. A002110, A006093, A039787, A053669, A249270, A356094 (denominators).
%Y A356093 Similar sequences: A038110, A338559, A340818, A341431, A342450, A342479.
%K A356093 nonn,frac
%O A356093 1,3
%A A356093 _Amiram Eldar_, Jul 26 2022