A039787 -id:A039787 - 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.

A075431 Primes of the form n+mu(n), where mu is the Moebius function (A008683).

Original entry on oeis.org

2, 7, 11, 23, 29, 41, 47, 59, 83, 101, 107, 109, 113, 137, 167, 173, 179, 181, 211, 227, 229, 257, 263, 281, 317, 331, 347, 353, 359, 373, 383, 401, 409, 433, 463, 467, 479, 503, 547, 563, 571, 587, 601, 617, 641, 653, 677, 691, 709, 719, 761, 821, 829, 839, 853

Offset: 1

Reinhard Zumkeller, Sep 15 2002

nonn

Subsequence of A075430.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A063015, A039787, A049097.

isok(p)={isprime(p) && (moebius(p+1) == -1 || moebius(p-1) == 1)} \\ Andrew Howroyd, Apr 20 2021

Terms a(41) and beyond from Andrew Howroyd, Apr 20 2021

A153214 Primes p such that p+-2 and p+-3 are not squarefree.

Original entry on oeis.org

47, 1447, 1847, 3701, 6653, 11273, 14947, 15727, 17053, 18493, 21661, 24923, 26647, 29153, 32789, 33023, 38873, 39323, 42437, 42923, 44053, 47527, 47977, 49853, 52027, 52153, 56747, 56873, 59929, 71147, 74189, 79427, 80953, 99277, 99713

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 20 2008

nonn

Cf. A049282, A049097, A039787, A075432, A153207, A153208, A153209, A153210, A153213

lst={};Do[p=Prime[n];If[ !SquareFreeQ[p-2]&&!SquareFreeQ[p+2]&&!SquareFreeQ[p-3]&&!SquareFreeQ[p+3],AppendTo[lst,p]],{n,3*7!}];lst
Select[Prime[Range[10000]],NoneTrue[#+{-3,-2,2,3},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 27 2019 *)

A124045 Numbers n such that n^2 divide A123269(n) = Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ].

Original entry on oeis.org

1, 2, 3, 6, 42

Offset: 1

Alexander Adamchuk, Nov 02 2006

A123269(n) = Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ] = {1, 28, 7625731729896, ...}. Numbers n that divide A123269(n) are listed in A124391(n) = {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 40, 42, 43, ...}.

Cf. A123269, A124391, A039787, A086787.

Do[f=Sum[Mod[Sum[Mod[Sum[PowerMod[i, j^k, n^2], {i, 1, n}], n^2], {j, 1, n}], n^2], {k, 1, n}]; If[IntegerQ[f/n^2], Print[n]], {n, 1, 103}]

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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A075431 Primes of the form n+mu(n), where mu is the Moebius function (A008683).

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A153214 Primes p such that p+-2 and p+-3 are not squarefree.

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A124045 Numbers n such that n^2 divide A123269(n) = Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ].

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