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A378620 Lesser prime index of twin primes with nonsquarefree mean.

%I A378620 #12 Jul 13 2025 16:35:55
%S A378620 2,5,7,17,20,28,35,41,43,45,49,52,57,64,69,81,83,98,109,120,140,144,
%T A378620 152,171,173,176,178,182,190,206,215,225,230,236,253,256,262,277,286,
%U A378620 294,296,302,307,315,318,323,336,346,373,377,390,395,405,428,430,444
%N A378620 Lesser prime index of twin primes with nonsquarefree mean.
%C A378620 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A378620 This is a subset of A029707 (twin prime indices). The other twin primes are A068361, so A029707 is the disjoint union of A068361 and A378620.
%H A378620 Harvey P. Dale, <a href="/A378620/b378620.txt">Table of n, a(n) for n = 1..1000</a>
%F A378620 prime(a(n)) = A061368(n).
%t A378620 Select[Range[100],Prime[#]+2==Prime[#+1]&&!SquareFreeQ[Prime[#]+1]&]
%t A378620 PrimePi/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&!SquareFreeQ[Mean[#]]&][[;;,1]] (* _Harvey P. Dale_, Jul 13 2025 *)
%Y A378620 The lesser of twin primes is A001359, index A029707 (complement A049579).
%Y A378620 The greater of twin primes is A006512, index A107770 (complement appears to be A168543).
%Y A378620 A subset of A029707 (twin prime lesser indices).
%Y A378620 Prime indices of the primes listed by A061368.
%Y A378620 Indices of twin primes with squarefree mean are A068361.
%Y A378620 A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
%Y A378620 A005117 lists the squarefree numbers, differences A076259.
%Y A378620 A006562 finds balanced primes.
%Y A378620 A013929 lists the nonsquarefree numbers, differences A078147.
%Y A378620 A014574 is the intersection of A006093 and A008864.
%Y A378620 A038664 finds the first position of a prime gap of 2n.
%Y A378620 A046933 counts composite numbers between primes.
%Y A378620 A120327 gives the least nonsquarefree number >= n.
%Y A378620 Cf. A007674, A072284, A068360.
%Y A378620 Cf. A057627, A061398, A061399, A067535, A070321, A071403, A112925.
%Y A378620 Cf. A000720, A025584, A067774, A122535, A155752, A179067.
%K A378620 nonn
%O A378620 1,1
%A A378620 _Gus Wiseman_, Dec 10 2024