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A360758 Numbers k for which k' - 1 and k' + 1 are twin primes, where the prime denotes the arithmetic derivative.

%I A360758 #15 Apr 01 2023 23:53:21
%S A360758 4,8,9,35,36,64,65,68,77,81,112,160,161,185,188,208,209,221,225,236,
%T A360758 335,341,371,377,428,437,441,485,515,576,596,611,671,707,731,736,756,
%U A360758 767,779,783,792,851,868,899,917,952,965,972,1007,1028,1067,1115,1152,1157
%N A360758 Numbers k for which k' - 1 and k' + 1 are twin primes, where the prime denotes the arithmetic derivative.
%C A360758 Numbers that have an arithmetic derivative equal to the average of twin prime pairs (A014574).
%C A360758 If p is in A022005 then m = 5*p is a term. Indeed, p is prime and m' = (5*p)' = p + 5 and m' - 1 = p + 4 and m' + 1 = p + 6 which are twin prime numbers.
%C A360758 If p is in A046138 then m = 7*p is a term. Indeed, p is prime and m' = (7*p)' = p + 7 and m' - 1 = p + 6 and m' + 1 = p + 8 which are twin prime numbers.
%C A360758 If p is in A212492 then m = 11*p is a term. Indeed, p is prime and m' = (11*p)' = p + 11 and m' - 1 = p + 10 and m' + 1 = p + 12 which are twin prime numbers.
%e A360758 4' =  4, 4' - 1 =  4 - 1 =  3, 4' + 1 =  4 + 1 =  5, so 4 is a term.
%e A360758 8' = 12, 8' - 1 = 12 - 1 = 11, 8' + 1 = 12 + 1 = 13, so 8 is a term.
%e A360758 9' =  6, 9' - 1 =  6 - 1 =  5, 9' + 1 =  6 + 1 =  7, so 9 is a term.
%t A360758 d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[1200], And @@ PrimeQ[d[#] + {-1, 1}] &] (* _Amiram Eldar_, Mar 01 2023 *)
%o A360758 (Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in  [1..#Factorisation(n)]])>; [p:p in [2..1200]| IsPrime(Floor(f(p))-1) and IsPrime(Floor(f(p))+1)];
%Y A360758 Cf. A003415, A014574, A022005, A046138, A212492, A077800.
%K A360758 nonn
%O A360758 1,1
%A A360758 _Marius A. Burtea_, Mar 01 2023