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

1,3

Comments

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... .

Links

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

Formula

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.

Example

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

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A058316 A082386 A336521 * A233308 A028306 A111259

Adjacent sequences: A356090 A356091 A356092 * A356094 A356095 A356096

Keyword

nonn,frac

Author

Amiram Eldar, Jul 26 2022

Status

approved