A334896 Read terms e = T(n,k) in A333624 as Product(prime(k)^e) for n in A334769.

648, 648, 686, 12096, 12096, 686, 192000, 139968, 192000, 139968, 1866240, 179712, 179712, 1866240, 814968, 2101248, 102036672, 331776000, 102036672, 331776000, 2101248, 814968, 179712000, 4423680000, 1866240000, 131010048, 179712000, 4423680000, 1866240000, 131010048

Offset: 1

Author

Michael De Vlieger, May 23 2020

Keywords

• nonn

Comments

Row a(n) of A067255 = row A334769(n) of A333624.

An XOR-triangle t(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit.

Let T(n,k) address the terms in the k-th position of row n in A333624.

This sequence encodes T(n,k) via A067255 to succinctly express the number of zero-triangles in A334769(n). To decode a(n) => A333624(A334769(n)), we use A067255(a(n)).

Examples

			a(1) = 648, since b(A334769(1)) = b(151) = 10010111, which generates T(151) as shown below, replacing 1 with "@" and 0 with ".":
  @ . . @ . @ @ @
   @ . @ @ @ . .
    @ @ . . @ .
     . @ . @ @
      @ @ @ .
       . . @
        . @
         @
In this figure we see 3 zero-triangles of side length k = 1, and 4 of side length k = 2, therefore, T(1,1) = 3 and T(1,2) = 4. This becomes 2^3 * 3^4 = 8 * 81 = 648.
Relationship of this sequence to A334556 and A333624:
        n  A334769(n)  a(n)  Row n of A333624
      --------------------------------------
       1    151       648    3, 4
       2    233       648    3, 4
       3    543       686    1, 0, 0, 3
       4    599     12096    6, 3, 0, 1
       5    937     12096    6, 3, 0, 1
       6    993       686    1, 0, 0, 3
       7   1379    192000    9, 1, 3
       8   1483    139968    6, 7
       9   1589    192000    9, 1, 3
      10   1693    139968    6, 7
      11   2359   1866240    9, 6, 1
      12   2391    179712    9, 3, 0, 0, 0, 1
      13   3753    179712    9, 3, 0, 0, 0, 1
      14   3785   1866240    9, 6, 1
      15   8607    814968    3, 3, 0, 3, 1
      16   9559   2101248   12, 3, 0, 0, 0, 0, 0, 1
      ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to XOR-triangles

Crossrefs

Cf. A038554, A067255, A070939, A333624, A334591, A334556, A334769.

Programs

Mathematica

With[{s = Rest[Import["https://oeis.org/A334769/b334769.txt", "Data"][[All, -1]]]}, Map[With[{w = NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &]}, If[Length@ # == 0, 1, Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #] &@ ReplacePart[ConstantArray[0, Max@ #[[All, 1]]], Map[#1 -> #2 & @@ # &, #]]] &@ Tally@ Flatten@ Array[If[# == 1, Map[If[First@ # == 1, Nothing, Length@ #] &, Split@ w[[#]]], Map[If[First@ # == -1, Length@ #, Nothing] &, Split[w[[#]] - Most@ w[[# - 1]]]]] &, Length@ w]] /. -Infinity -> 0 &, s[[1 ;; 29]]]]