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The maximum exponent in prime factorization of the average of n-th twin prime pair.
2

%I #17 Sep 09 2024 02:41:03

%S 2,1,2,2,1,1,2,3,1,3,1,2,2,6,2,2,4,3,1,3,2,2,4,1,2,1,3,1,1,2,4,1,2,1,

%T 2,2,3,2,2,2,7,1,2,1,1,3,2,2,1,4,3,4,2,1,1,2,4,1,2,2,3,2,1,3,6,1,2,1,

%U 4,1,2,1,2,5,1,7,3,1,2,1,3,3,4,5,2,2,2,2,5,2,1,1,2,3,2,1,2,3,3,2,2,1,1,1,2

%N The maximum exponent in prime factorization of the average of n-th twin prime pair.

%H Amiram Eldar, <a href="/A078573/b078573.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A051903(A014574(n)).

%e 10th twin prime pair = (A001359(10), A006512(10)) = (107,109), hence A014574(10) = 108 = 2^2 * 3^2, therefore a(10) = 3.

%t midQ[n_] := PrimeQ[n-1] && PrimeQ[n+1]; f[n_] := Max @@ FactorInteger[n][[;;,2]]; f /@ Select[Range[5000], midQ] (* _Amiram Eldar_, Nov 03 2019 *)

%t Max[FactorInteger[Mean[#]][[All,2]]]&/@Select[Partition[Prime[Range[1000]],2,1],#[[2]] - #[[1]]==2&] (* _Harvey P. Dale_, Apr 21 2022 *)

%o (PARI) lista(pmax) = {my(prv = 2); forprime(p = 3, pmax, if(p - prv ==2, print1(vecmax(factor(p-1)[,2]), ", ")); prv = p);} \\ _Amiram Eldar_, Sep 09 2024

%Y Cf. A001359, A006512, A051903, A014574, A078570, A078572.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Nov 29 2002