A071318 - OEIS

%I #36 Jan 05 2024 02:56:29

%S 44,49,75,98,99,116,147,171,244,260,275,315,332,363,387,475,476,507,

%T 524,531,548,549,603,604,636,692,724,725,747,764,774,819,844,845,846,

%U 867,908,924,931,963,980,1035,1075,1083,1179,1196,1251,1274,1275,1324

%N Lesser of 2 consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that both k and k+1 are in A067259.

%C The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 5, 41, 407, 4125, 41215, 412331, 4123625, 41236308, ... . Apparently, the asymptotic density of this sequence exists and equals 0.041236... . - _Amiram Eldar_, Jan 18 2023

%C The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^3) - 2 * Product_{p prime} (1 - 1/p^2 - 1/p^3) + Product_{p prime} (1 - 2/p^2) = 0.041236147082334172926... . - _Amiram Eldar_, Jan 05 2024

%H Amiram Eldar, <a href="/A071318/b071318.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%F A051903(k) = A051903(k+1) = 2 when k is a term.

%e 75 is a term since 75 = 3*5^2 and 76 = 2^2*19.

%t With[{s = Select[Range[1350], And[MemberQ[#, 2], FreeQ[#, k_ /; k > 2]] &@ FactorInteger[#][[All, -1]] &]}, Function[t, Part[s, #] &@ Position[t, 1][[All, 1]]]@ Differences@ s] (* _Michael De Vlieger_, Jul 30 2017 *)

%o (Haskell)

%o a071318 n = a071318_list !! (n-1)

%o a071318_list = [x | x <- [1..],  a212793 x == 1, a008966 x == 0,

%o                     let y = x+1, a212793 y == 1, a008966 y == 0]

%o -- _Reinhard Zumkeller_, May 27 2012

%o (PARI) isok(n) = (n>1) && (vecmax(factor(n)[, 2])==2) && (vecmax(factor(n+1)[, 2])==2); \\ _Michel Marcus_, Aug 02 2017

%Y Subsequence of A067259, A340152.

%Y Cf. A007674, A063528, A071619, A071620, A071621, A071622, A071623.

%Y Cf. A008966, A051903, A212793.

%K nonn

%O 1,1

%A _Labos Elemer_, May 29 2002