A122692 Cubeful numbers whose neighbors are also cubeful.

1376, 4375, 4913, 5751, 6859, 13311, 13376, 16120, 21249, 22625, 22626, 24353, 25624, 28376, 31375, 32751, 33615, 40473, 41743, 48249, 49625, 49735, 52624, 55376, 57968, 58375, 59751, 75249, 76625, 79624, 82376, 85375, 86751, 90208

Offset: 1

Author

Tanya Khovanova, Oct 21 2006

Keywords

• nonn

Comments

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

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale, terms 1001..3903 from Robert Israel)

Crossrefs

Subsequence of A046099 and A068140.

Cf. A088453, A340153.

Programs

Maple

N := 10^6: # get all terms <= N
CF := {seq(seq(x^3 * y, y = 1..floor(N/x^3)), x = 2..floor(N^(1/3)))}:
CF intersect map(`-`, CF, 1) intersect map(`+`, CF, 1): # Robert Israel, Jul 16 2014

Mathematica

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

PARI

iscubefree(n) = vecsort(factor(n)~, 2, 4)[2, 1] < 3
s = []; for(n = 3, 200000, if(!iscubefree(n - 1) && !iscubefree(n) && !iscubefree(n + 1), s = concat(s, n))); s \\ Colin Barker, Jul 16 2014

PARI

A051903(n)=if(n>1, vecmax(factor(n)[, 2]), 0)
is(n)=A051903(n)>2 && A051903(n-1)>2 && A051903(n+1)>2 \\ Charles R Greathouse IV, Jul 23 2014