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A294731 Smallest average of a twin prime pair divisible by the n-th prime, i.e. A090530(n), divided by 6*prime(n).

%I A294731 #20 Aug 25 2025 09:23:11
%S A294731 1,1,3,4,1,2,1,2,7,9,5,4,1,20,3,43,4,3,14,22,9,8,19,7,1,1,8,4,24,5,1,
%T A294731 2,2,13,4,6,5,9,22,3,15,6,11,3,7,5,20,5,6,7,3,3,9,14,10,2,35,2,1,10,
%U A294731 25,17,1,35,5,4,1,18,15,12,25,1,2,5
%N A294731 Smallest average of a twin prime pair divisible by the n-th prime, i.e. A090530(n), divided by 6*prime(n).
%C A294731 The sequence starts at n=3, because A090530(1)=4 is not divisible by 6*2 and A090530(2)=6 is not divisible by 6*3.
%C A294731 The positions of ones in the sequence are given by A060212, i.e. a(A000720(A060212(n)))=1 for all n>=3.
%H A294731 Hugo Pfoertner, <a href="/A294731/b294731.txt">Table of n, a(n) for n = 3..10000</a>
%F A294731 a(n) = A090530(n) / ( 6 * prime(n) ) for n >= 3.
%F A294731 a(n) = A071407(n) / 6. - _Amiram Eldar_, Aug 25 2025
%e A294731 a(5)=3 because 198 is the smallest average of a twin prime pair {197,199} that is divisible by the 5th prime 11: 3 = 198 / (6*11).
%t A294731 a[n_] := Module[{p = Prime[n], k = 1}, While[! PrimeQ[6*k*p - 1] || ! PrimeQ[6*k*p + 1], k++]; k]; Array[a, 100, 3] (* _Amiram Eldar_, Aug 25 2025 *)
%Y A294731 Cf. A014574, A060212, A071407, A090530, A182481.
%K A294731 nonn,changed
%O A294731 3,3
%A A294731 _Hugo Pfoertner_, Nov 09 2017