cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A264164 3-smooth numbers whose number of divisors is not 3-smooth.

Original entry on oeis.org

16, 48, 64, 81, 144, 162, 192, 324, 432, 512, 576, 648, 729, 1024, 1296, 1458, 1536, 1728, 2592, 2916, 3072, 3888, 4096, 4608, 5184, 5832, 8192, 9216, 10368, 11664, 12288, 13824, 15552, 16384, 19683, 20736, 23328, 24576, 27648, 34992, 36864, 39366, 41472
Offset: 1

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Author

Reinhard Zumkeller, Nov 19 2015

Keywords

Examples

			a(12) = 648 = 2^3*3^4 = A003586(36) and A000005(648) = 20 = 2^2*5.
a(13) = 729 = 3^6 = A003586(37) and A000005(729) = 7.
A003586(38) = 768 = 2^8*3 is not a term, as A000005(768) = 18 = 2*3^2.
		

Crossrefs

Cf. A000005, A003586, A065333, A069353, A264165 (complement with respect to A003586).

Programs

  • Haskell
    a264164 n = a264164_list !! (n-1)
    a264164_list = filter ((== 0) . a065333 . a000005') a003586_list
  • Mathematica
    smQ[n_] := n == Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]);
    seq[max_] := Sort@ Flatten@ Table[If[smQ[i + 1] && smQ[j + 1], Nothing, 2^i * 3^j], {i, 0 , Log2[max]}, {j, 0, Log[3, max/2^i]}]; seq[42000] (* Amiram Eldar, Sep 03 2023 *)

Formula

A065333(a(n)) * (1 - A065333(A000005(a(n)))) = 1.
Sum_{n>=1} 1/a(n) = 3 - (Sum_{n>=1} 1/2^A069353(n)) * (Sum_{n>=1} 1/3^A069353(n)) = 0.14870906483739045152... . - Amiram Eldar, Apr 17 2025
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