A356371 -id:A356371 - cp's OEIS frontend

A358127 a(n) is the cardinality of the set of pairwise gcd's of {prime(1)+1, ..., prime(n)+1}.

1, 3, 4, 5, 5, 5, 5, 7, 8, 8, 8, 9, 9, 11, 12, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 18, 19, 20, 21, 22, 22, 23, 23, 23, 23, 23, 24, 24, 26, 27, 29, 29, 30, 32, 32, 33, 35, 36, 36, 37, 37, 37, 37, 38, 38, 39, 39, 39, 39, 40, 40, 42, 42, 43, 43, 43, 44, 45, 45, 48, 48, 48, 48, 50, 50, 50, 50

Offset: 2

Gleb Ivanov, Oct 30 2022

nonn

			For n = 3 initial set is {2+1, 3+1, 5+1} and after applying gcd for each distinct pair of elements we get {1, 2, 3} set with cardinality of a(3) = 3.

Cf. A008864, A356371, A214799.

from sympy import nextprime
from math import gcd
from itertools import combinations
pr, terms = [2,3], []
for i in range(100):
    terms.append(len(set([gcd(t[0]+1, t[1]+1) for t in combinations(pr,2)])))
    pr.append(nextprime(pr[-1]))
print(terms)

from math import gcd
from itertools import count, islice
from sympy import prime
def A358127_gen(): # generator of terms
    a, b = [3], set()
    for n in count(2):
        q = prime(n)+1
        b |= set(gcd(p,q) for p in a)
        yield len(b)
        a.append(q)
A358127_list = list(islice(A358127_gen(),100)) # Chai Wah Wu, Nov 02 2022

A358178 a(n) is the cardinality of the set of distinct pairwise gcd's of {1! + 1, ..., n! + 1}.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 11, 12, 12, 12, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18

Offset: 1

Gleb Ivanov, Nov 02 2022

nonn

			For n = 6 initial set is {1+1, 2+1, 6+1, 24+1, 120+1, 720+1} and after applying gcd for each distinct pair of elements we get {1, 7} set with cardinality of a(6) = 2.

Cf. A038507, A358127, A356371, A214799.

from math import gcd, factorial
from itertools import combinations
f, terms = [2,], []
for i in range(2,100):
    f.append(factorial(i)+1)
    terms.append(len(set([gcd(*t) for t in combinations(f, 2)])))
print(terms)

from math import gcd
from itertools import count, islice
def A358178_gen(): # generator of terms
    m, f, g = 1, [], set()
    for n in count(1):
        m *= n
        g |= set(gcd(d,m+1) for d in f)
        f.append(m+1)
        yield len(g)
A358178_list = list(islice(A358178_gen(),20)) # Chai Wah Wu, Dec 15 2022

cp's OEIS Frontend

A358127 a(n) is the cardinality of the set of pairwise gcd's of {prime(1)+1, ..., prime(n)+1}.

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A358178 a(n) is the cardinality of the set of distinct pairwise gcd's of {1! + 1, ..., n! + 1}.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Python