A122692 - OEIS

%I #30 Sep 12 2024 07:47:46

%S 1376,4375,4913,5751,6859,13311,13376,16120,21249,22625,22626,24353,

%T 25624,28376,31375,32751,33615,40473,41743,48249,49625,49735,52624,

%U 55376,57968,58375,59751,75249,76625,79624,82376,85375,86751,90208

%N Cubeful numbers whose neighbors are also cubeful.

%C The asymptotic density of this sequence is 1 - 3/zeta(3) + 3 * Product_{p prime} (1 - 2/p^3) - Product_{p prime} (1 - 3/p^3) = 1 - 3 * A088453 + 3 * A340153 - Product_{p prime} (1 - 3/p^3) = 0.00038922120241968636455... . - _Amiram Eldar_, Sep 12 2024

%H Amiram Eldar, <a href="/A122692/b122692.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale, terms 1001..3903 from Robert Israel)

%e 1376 is divisible by 8, and its neighbors 1375 and 1377 are divisible by 125 and 27, respectively.

%p N := 10^6: # get all terms <= N

%p CF := {seq(seq(x^3 * y, y = 1..floor(N/x^3)), x = 2..floor(N^(1/3)))}:

%p CF intersect map(`-`, CF, 1) intersect map(`+`, CF, 1): # _Robert Israel_, Jul 16 2014

%t Select[Range[2, 100000], Max[Transpose[FactorInteger[ # ]][[2]]] >= 3 && Max[Transpose[FactorInteger[# + 1]][[2]]] >= 3 && Max[Transpose[FactorInteger[# - 1]][[2]]] >= 3 &]

%t cnQ[{a_,b_,c_}] := And@@(# > 2 &/@{a, b, c}); Flatten[Position[Partition[Table[Max[Transpose[FactorInteger[n]][[2]]], {n, 91000}], 3, 1], _?(cnQ[#] &)]] + 1 (* _Harvey P. Dale_, Jul 28 2013 *)

%o (PARI)

%o iscubefree(n) = vecsort(factor(n)~, 2, 4)[2, 1] < 3

%o s = []; for(n = 3, 200000, if(!iscubefree(n - 1) && !iscubefree(n) && !iscubefree(n + 1), s = concat(s, n))); s \\ _Colin Barker_, Jul 16 2014

%o (PARI) A051903(n)=if(n>1, vecmax(factor(n)[, 2]), 0)

%o is(n)=A051903(n)>2 && A051903(n-1)>2 && A051903(n+1)>2 \\ _Charles R Greathouse IV_, Jul 23 2014

%Y Subsequence of A046099 and A068140.

%Y Cf. A088453, A340153.

%K nonn

%O 1,1

%A _Tanya Khovanova_, Oct 21 2006