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A387465 Irregular triangle T(n,k), n >= 0, 0 <= k < 2^(n-1), where T(n,k) = Product_{j=0..n-1} prime(n-j)^((j+1)*d_j), where d_j is the bit with digit weight 2^j in the binary expansion of 2^(n-1)+k.

%I A387465 #18 Sep 08 2025 23:37:06
%S A387465 1,2,4,12,8,40,72,360,16,112,400,2800,432,3024,10800,75600,32,352,
%T A387465 1568,17248,4000,44000,196000,2156000,2592,28512,127008,1397088,
%U A387465 324000,3564000,15876000,174636000,64,832,7744,100672,21952,285376,2656192,34530496,40000,520000
%N A387465 Irregular triangle T(n,k), n >= 0, 0 <= k < 2^(n-1), where T(n,k) = Product_{j=0..n-1} prime(n-j)^((j+1)*d_j), where d_j is the bit with digit weight 2^j in the binary expansion of 2^(n-1)+k.
%C A387465 For n >= 1, row n consists of the even unitary divisors of A006939(n).
%C A387465 The range of properties is best understood when also viewed as a linear sequence with offset 0, so that a(floor(2^(n-1))+k) = T(n,k).
%C A387465 For even n > 0, a(n) is powerful.
%C A387465 For odd n > 1, a(n) is in A332785.
%C A387465 a(1) = 2 is the only prime term.
%C A387465 a(0) = 1 and a(1) = 2 are the only squarefree terms.
%C A387465 a(2^k) = 2^(k+1).
%C A387465 Perfect powers in this sequence include A000079, but also numbers like 400 = prime(1)^4 * prime(3)^2.
%H A387465 Michael De Vlieger, <a href="/A387465/b387465.txt">Table of n, a(n) for n = 0..16384</a>
%H A387465 Michael De Vlieger, <a href="/A387465/a387465.png">Log log scatterplot of a(n)</a> n = 0..2^14.
%H A387465 Michael De Vlieger, <a href="/A387465/a387465_1.png">Plot prime(i)^j at (x,y) = (n,i)</a>, n = 0..2047, 16X vertical exaggeration, with a color function representing j = 1 in black, j = 2 in red, j = 3 in orange, ..., j = 15 in magenta.
%F A387465 Let row_index(n) = A029837(n+1) = ceiling(log_2(n+1)), giving the row of the triangle that contains term a(n) of the linear sequence.
%F A387465 Let lin_index(n,k) = floor(2^(n-1))+k, giving the index in the linear sequence corresponding to term T(n,k) of the triangle.
%F A387465 a(0) = 1; otherwise:
%F A387465   a(2n) = A064549(a(n)) = a(n) * rad(a(n));
%F A387465   a(2n+1) = a(2n) * prime(row_index(n)).
%F A387465 T(n,0) = 2^n.
%F A387465 T(n,k) = T(n,0) * A242378(n - row_index(k), a(k)), where A242378(i,j) "adds i to the indices of the prime factors of j".
%F A387465 T(n,ceiling(2^(n-1))-1) = A006939(n).
%F A387465 For n > 1, T(n,1) = A265127(n) = 2^n * prime(n).
%F A387465 For n > 2, T(n,2^(n-1)) = A167747(n) = phi(6^n) = 2^n * 3^(n-1).
%F A387465 For n > 2, T(n,2^(n-2)) = A093141(n-1) = 4 * 10^(n-1). = 2^n * 5^(n-2).
%F A387465 T(n,k) = max({j >= 1 : j|A006939(n) and gcd(j, A019565(A036044(lin_index(n,k)))) = 1}).
%F A387465 A110765(n) = A007947(a(n)).
%F A387465 A001221(a(n)) = A000120(n).
%F A387465 A001222(a(n)) = A029931(n).
%e A387465 Table begins:
%e A387465 n\k  0    1    2     3    4     5      6       7
%e A387465 ------------------------------------------------
%e A387465 0:   1;
%e A387465 1:   2;
%e A387465 2:   4,  12;
%e A387465 3:   8,  40,  72,  360;
%e A387465 4:  16, 112, 400, 2800, 432, 3024, 10800, 75600;
%e A387465      ...
%e A387465 Table showing prime power decomposition of a(n), where A067255(a(n)) represents prime(i)^j | a(n), with j in the i-th position, replacing 0 with "." for visibility:
%e A387465  n     a(n)  A067255(a(n))
%e A387465 --------------------------
%e A387465  0       1   .
%e A387465  1       2   1
%e A387465  2       4   2
%e A387465  3      12   21
%e A387465  4       8   3
%e A387465  5      40   3.1
%e A387465  6      72   32
%e A387465  7     360   321
%e A387465  8      16   4
%e A387465  9     112   4..1
%e A387465 10     400   4.2
%e A387465 11    2800   4.21
%e A387465 12     432   43
%e A387465 13    3024   43.1
%e A387465 14   10800   432
%e A387465 15   75600   4321
%t A387465 f[x_] := If[x == 1, {0},
%t A387465  Function[g,ReplacePart[Table[0, {PrimePi[f[[-1, 1]] ]}], #] &@
%t A387465      Map[PrimePi@ First@ # -> Last@ # &, g] ]@ FactorInteger@ x];
%t A387465 Table[f[Reverse@ Range[Length[#]]*#] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 120}]
%Y A387465 Cf. A001597, A001694, A110765 (squarefree kernel), A286708, A362227, A363250, A384003.
%Y A387465 All terms are in A304686.
%Y A387465 See the comments for the relationships with A000079, A006939, A332785.
%Y A387465 See the formula section for the relationships with A000120, A001221, A001222, A007947, A019565, A029837, A029931, A036044, A064549, A093141, A167747, A242378, A265127.
%Y A387465 See the examples for the relationship with A067255.
%K A387465 nonn,tabf,easy,new
%O A387465 0,2
%A A387465 _Michael De Vlieger_ and _Peter Munn_, Aug 29 2025