A070195 -id:A070195 - cp's OEIS frontend

A335200 Positions of difference 12 in A070195.

2, 14, 31, 47, 78, 83, 229, 242, 278, 294, 312, 341, 345, 350, 381, 401, 429, 441, 476, 617, 783, 796, 800, 812, 825, 849, 911, 921, 931, 940, 1054, 1087, 1103, 1142, 1146, 1210, 1291, 1340, 1391, 1397, 1447, 1556, 1594, 1804, 1828, 1855, 1857, 1877, 1953

Offset: 1

Zak Seidov, May 26 2020

nonn

12 is the minimal difference in A070195.

			A070195(2) = A070195(3) - 12, or 30 = 42 - 12, so 2 is a term.
A070195(14) = A070195(15) - 12, or 1290 = 1302 - 12, so 14 is a term.

Cf. A070195.

Position[Differences @ Select[Range[10^6], SquareFreeQ[#] && And @@ PrimeQ[# + {-1, 1}] &], 12] //Flatten (* Amiram Eldar, May 27 2020 *)

A078579 Squarefree kernel of the average of n-th twin prime pair.

Original entry on oeis.org

2, 6, 6, 6, 30, 42, 30, 6, 102, 6, 138, 30, 30, 6, 66, 114, 30, 30, 282, 78, 174, 210, 6, 462, 174, 570, 30, 618, 642, 330, 30, 822, 138, 858, 42, 510, 258, 210, 354, 546, 6, 1230, 426, 1290, 1302, 330, 714, 66, 1482, 186, 402, 30, 834, 1698, 1722, 894, 78

Offset: 1

Reinhard Zumkeller, Nov 29 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007947, A014574, A070195, A078580.

midQ[n_] := PrimeQ[n-1] && PrimeQ[n+1]; f[n_] := Times @@ FactorInteger[n][[;;,1]]; f /@ Select[Range[2000], midQ] (* Amiram Eldar, Nov 03 2019 *)

a(n) = A007947(A014574(n)).

a(A070195(n)) = A070195(n).

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

Original entry on oeis.org

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

Amiram Eldar, Jun 13 2022

			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

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

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

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

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

cp's OEIS Frontend

A335200 Positions of difference 12 in A070195.

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A078579 Squarefree kernel of the average of n-th twin prime pair.

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

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Mathematica

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Formula