A356094 -id:A356094 - cp's OEIS frontend

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

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, 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, 1, 4, 2, 1, 8, 1, 2, 16, 1, 1, 2, 9, 1, 2, 6, 40, 4, 1, 2, 1

Offset: 1

Amiram Eldar, Jul 26 2022

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.

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... .

			Fractions begin with 1/2, 1/3, 2/15, 1/35, 1/231, 2/5005, 8/255255, 3/1616615, 1/10140585, 2/462120945, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002110, A006093, A039787, A053669, A249270, A356094 (denominators).

Similar sequences: A038110, A338559, A340818, A341431, A342450, A342479.

primorial[n_] := Product[Prime[i], {i, 1, n}]; Numerator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 100}]]

a(n) = numerator((prime(n)-1)/factorback(primes(n))); \\ Michel Marcus, Jul 26 2022

from math import gcd
from sympy import prime, primorial
def A356093(n): return (p:=prime(n)-1)//gcd(p,primorial(n)) # Chai Wah Wu, Jul 26 2022

a(n) = 1 iff prime(n) is in A039787.
Let f(n) = a(n)/A356094(n):
f(n) = A006093(n)/A002110(n).
Sum_{n>=1} f(n) = 1.
Sum_{n>=1} f(n) * prime(n) = A249270.

cp's OEIS Frontend

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

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