A354950 The number of squarefree numbers whose largest prime divisor is prime(n) and that are averages of twin prime pairs.

0, 1, 1, 1, 2, 2, 3, 7, 4, 9, 20, 31, 57, 88, 139, 282, 421, 806, 1397, 2572, 4440, 7863, 14580, 26211, 47727, 86929, 159972, 292650, 542477, 1000087, 1850347, 3432551, 6381199

Offset

1,5

Links

Table of n, a(n) for n=1..33.

Formula

Conjecture: Limit_{n->oo} log(a(n))/(n*log(n)) = c ~ 0.13... .

Example

n  prime(n)  a(n)  terms k of A070195 with A006530(k) = prime(n)
-  --------  ----  ---------------------------------------------
1   2        0     -
2   3        1     6
3   5        1     30
4   7        1     42
5  11        2     462, 2310
6  13        2     858, 2730
7  17        3     102, 9282, 102102
8  19        7     570, 1482, 6270, 21318, 43890, 51870, 1939938

Mathematica

a[n_] := Count[Prime[n] * Divisors[Product[Prime[i], {i, 1, n - 1}]], _?(PrimeQ[# - 1] && PrimeQ[# + 1] &)]; Array[a, 10]

Prog

(Python)
from math import prod
from itertools import combinations
from sympy import primerange, prime, isprime
def A354950(n):
    plist = list(primerange(2, p:=prime(n)))
    return sum(1 for l in range(1, n) for d in combinations(plist, l) if isprime((q:= prod(d)*p)-1) and isprime(q+1)) # Chai Wah Wu, Jun 14 2022

Crossrefs

Cf. A005117, A006530, A014574, A070195, A354951.

Sequence in context: A210753 A208762 A209746 * A267822 A210598 A330728

Adjacent sequences: A354947 A354948 A354949 * A354951 A354952 A354953

Keyword

nonn,more

Author

Amiram Eldar, Jun 13 2022

Status

approved