A235483 - OEIS

%I #31 Aug 26 2019 12:50:51

%S 30,60,150,180,240,270,420,570,600,660,810,858,1020,1050,1230,1290,

%T 1320,1620,1950,2130,2310,2340,2550,2730,2790,2970,3000,3120,3300,

%U 3330,3360,3390,3540,3930,4020,4050,4230,4260,4650,4800,5010,5100,5280,5520,5640,5850

%N Average of twin prime pairs that contains at least one twin prime pair as divisors.

%C The corresponding numbers of twin prime pairs divisors are 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, ...

%C The subsequence with two twin prime pairs is {420, 1050, 2310, 2730, 3360, 5880, 6090, 6300, 7350, 7560, 8820, 9240, 10500, 10710, 11550, 11970, 15330, 18060, 19380, 21840, 25410, 26250, 26880, 28350, 29400, ...}.

%C The subsequence with three twin prime pairs is interesting because we find a majority of symmetrical numbers of the form abcabc or abcabc0 (see A235716). The subsequence with three twin prime pairs is 180180, 270270, 300300, 330330, 390390, 420420, 474810, 540540, 660660, 755790, 840840, 1231230, 1261260, 1501500, 1510320, 1621620, 1699110, ...

%C The subsequence with four twin prime pairs is 106696590, 107987880, 232792560, 244120170, 300690390, 358888530, 432311880, 466936470, 601380780, 670770870, 766275510, 892371480, 978498570, 998887890, 999068070, ... [_Michel Marcus_, Jan 11 2014]

%C Apparently most terms are congruent to 0 mod 30. Terms that are not: 858, 9438, 21318, 40698, 60918, 64782, 69498, 70122, 99528, ... - _Amiram Eldar_, Aug 26 2019

%H Amiram Eldar, <a href="/A235483/b235483.txt">Table of n, a(n) for n = 1..10000</a>

%e 30 is in the sequence because the prime divisors are 2, 3, 5 and {3,5} is a twin primes.

%e 420 is in the sequence because the prime divisors are 2, 3, 5, 7 => {3,5} and {5,7} are twin primes.

%p with(numtheory) :

%p      for n from 1 to 3400 do:

%p          p1:=ithprime(n):p2:=ithprime(n+1):

%p            if p2=p1+2

%p            then

%p            ii:=0:x:=factorset(p1+1):n1:=nops(x):

%p         for i from 1 to n1-1 do:

%p            if x[i+1]=x[i]+2

%p            then

%p            ii:=ii+1:

%p            else fi:

%p         od:

%p            if ii>0

%p            then

%p            printf(`%d, `,p1+1):

%p            else fi:

%p            fi:

%p        od:

%t Mean /@ Select[Split[Prime[Range[1000]], #2 - #1 == 2&], Length[#] == 2 && MemberQ[Differences[FactorInteger[Mean[#]][[All, 1]]], 2]&] (* _Jean-François Alcover_, Nov 06 2016 *)

%Y Cf. A001097, A014574, A235716.

%K nonn

%O 1,1

%A _Michel Lagneau_, Jan 11 2014