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
Examples
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
Programs
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Mathematica
a[n_] := Count[Prime[n] * Divisors[Product[Prime[i], {i, 1, n - 1}]], _?(PrimeQ[# - 1] && PrimeQ[# + 1] &)]; Array[a, 10] -
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
Formula
Conjecture: Limit_{n->oo} log(a(n))/(n*log(n)) = c ~ 0.13... .