A100380 -id:A100380 - cp's OEIS frontend

A265109 a(n) = largest k such that prime(n) + A002110(k) is prime.

0, 1, 2, 3, 3, 4, 6, 7, 8, 6, 5, 9, 8, 8, 14, 9, 16, 17, 14, 14, 16, 21, 11, 19, 14, 24, 25, 15, 18, 11, 28, 12, 8, 19, 16, 22, 35, 31, 36, 25, 31, 16, 40, 30, 23, 41, 39, 35, 10, 32, 43, 38, 24, 41, 19, 35, 23, 55, 54, 24, 53, 50, 57, 62, 48, 36, 64, 21, 45, 54

Offset: 1

Altug Alkan, Dec 01 2015

nonn

a(n) = n-1 iff a(n) is in A035346 for n > 1.

			a(4) = 3 because A002110(3) + prime(4) = A002110(3) + 7 = 37 is prime.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A000040, A002110, A035346, A100380.

a(n) = {my(k=1); while(k, if(ispseudoprime(prod(i=1, n-k, prime(i)) + prime(n)), return(n-k)); k++)}

from itertools import count
from sympy import isprime, prime, primorial
def A002110(n): return primorial(n) if n > 0 else 1
def A265109(n):
    pn = prime(n)
    return next(k for k in range(n-1, -1, -1) if isprime(pn+A002110(k)))
print([A265109(n) for n in range(1, 31)]) # Michael S. Branicky, Jan 10 2025

a(n) < n.

A356741 a(n) is the least prime(m) such that prime(n) + prime(m)# is prime, where prime(m)# denotes the product of the first m primes, or -1 if no such prime(m) exists.

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 7, 3, 2, 3, 3, 2, 5, 3, 3, 2, 3, 3, 2, 3, 5, 3, 11, 3, 2, 3, 2, 5, 11, 5, 3, 2, 7, 2, 3, 3, 5, 3, 3, 2, 5, 2, 3, 2, 5, 5, 3, 2, 7, 3, 2, 5, 3, 3, 3, 2, 3, 3, 2, 5, 7, 3, 2, 7, 5, 3, 5, 2, 5, 3, 5, 3, 3, 5, 3, 5, 7, 5, 5, 2, 7, 2, 3, 11, 3, 5, 3

Offset: 2

Alain Rocchelli, Sep 04 2022

nonn

Conjecture: Such a prime(m) exists for every n, i.e., a(n) is never -1 for n>1.

Conjecture: Limit_{N->oo} (Sum_{n=2..N} a(n)) / (Sum_{n=2..N} log(prime(n))) = C with C constant between 0.5 and 1 inclusive.

			For n=4, prime(4)=7, and m=1 gives prime(m)=2 and prime(n) + prime(m)# = 7 + 2 = 9 (nonprime), but m=2 gives prime(m)=3 and prime(n) + prime(m)# = 7 + 2*3 = 13 (prime), so a(4) = prime(2) = 3.

Alain Rocchelli, Table of n, a(n) for n = 2..10000

Cf. A002110, A100380.

a(n) = my(p=2, pr=2, pn=prime(n)); while (!isprime(pn+pr), p=nextprime(p+1); pr *= p); p; \\ Michel Marcus, Sep 05 2022

from sympy import isprime, nextprime, prime
def a(n):
    pn, pm, pmsharp = prime(n), 2, 2
    while not isprime(pn + pmsharp): pm = nextprime(pm); pmsharp *= pm
    return pm
print([a(n) for n in range(2, 89)]) # Michael S. Branicky, Sep 04 2022

a(n) = prime(A100380(n)). - Michel Marcus, Sep 12 2022

A380026 a(n) is the smallest prime p such that p - a(n-1) is a primorial, starting with a(1)=2.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 229, 439, 2749, 5059, 7369, 9679, 39709, 42019, 6469735249, 5766152219975951659023630035336134306565384015606066326325804059, 5766152219975951659023630035336134306565384015606073747063938869, 5766152219975951659023630035336134306565384015606073747287031739

Offset: 1

Hayden Chesnut, Jan 09 2025

nonn

			a(4) = 7
For primes greater than 7:
11 - 7 = 4 is not in A002110
13 - 7 = 6 is in A002110 so a(5) = 13

Cf. A000720, A002110, A100380, A380027.

from itertools import count, islice
from sympy import isprime, primorial
def A002110(n): return primorial(n) if n > 0 else 1
def agen(an=2): # generator of terms
    while True:
        yield an
        an = next(s for k in count(0) if isprime(s:=an+A002110(k)))
print(list(islice(agen(), 18))) # Michael S. Branicky, Jan 10 2025

from sympy import isprime
import primesieve
it = primesieve.Iterator()
chain = [2]
pchain = []
n = 1
while len(chain) < 18:
    while True:
        p = it.next_prime()
        if isprime(chain[-1]+n):
            chain.append(chain[-1]+n)
            print(len(chain))
            break
        n *= p
    p = it.skipto(0)
    n = 1
print(chain) # Hayden Chesnut, Jan 10 2025

a(n) = a(n-1) + A002110(A100380(A000720(a(n-1)))), for n > 1. - Michael S. Branicky, Jan 10 2025

cp's OEIS Frontend

A265109 a(n) = largest k such that prime(n) + A002110(k) is prime.

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A356741 a(n) is the least prime(m) such that prime(n) + prime(m)# is prime, where prime(m)# denotes the product of the first m primes, or -1 if no such prime(m) exists.

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A380026 a(n) is the smallest prime p such that p - a(n-1) is a primorial, starting with a(1)=2.

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Python

Python

Formula