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A154494 a(n+1) -+ a(n) = prime, a(n+1)*a(n) = average of twin prime pairs, a(1)=1, a(2)=6.

%I A154494 #9 Nov 29 2023 18:53:36
%S A154494 1,6,17,84,247,300,341,720,767,834,913,960,989,1590,1741,2772,2945,
%T A154494 3102,3451,3540,3563,4020,4159,4812,5431,5700,6131,6192,6245,8586,
%U A154494 9373,10056,10163,10314,10543,10944,11033,11844,12055,13566,14015,15048,15155
%N A154494 a(n+1) -+ a(n) = prime, a(n+1)*a(n) = average of twin prime pairs, a(1)=1, a(2)=6.
%C A154494 Sum and difference of any of two consecutive numbers in current sequence are prime numbers and multiplication and any two consecutive numbers is Average of twin prime pairs : 6-1=5; 6+1=7; 6*1=6, 17-6=11; 17+6=23; 17*6=102, ... .
%H A154494 Robert Israel, <a href="/A154494/b154494.txt">Table of n, a(n) for n = 1..10000</a>
%p A154494 f:= proc(x) local k;
%p A154494   for k from x+1 by 2 do
%p A154494     if isprime(k+x) and isprime(k-x) and isprime(k*x-1) and isprime(k*x+1) then return k fi
%p A154494   od;
%p A154494 end proc:
%p A154494 R:= 1,6: x:= 6:
%p A154494 for count from 3 to 100 do
%p A154494   x:= f(x);
%p A154494   R:= R,x
%p A154494 od:
%p A154494 R; # _Robert Israel_, Nov 29 2023
%t A154494 a=1;b=6;lst={a,b};Do[If[PrimeQ[n-b]&&PrimeQ[n+b]&&PrimeQ[n*b-1]&&PrimeQ[n*b+1],AppendTo[lst,n];a=b;b=n],{n,b+1,9!}];lst
%Y A154494 Cf. A154484, A154485, A154486, A154487, A154488, A154493.
%K A154494 nonn
%O A154494 1,2
%A A154494 _Vladimir Joseph Stephan Orlovsky_, Jan 10 2009
%E A154494 NAME adapted to offset. - _R. J. Mathar_, Jun 19 2021