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.

1, 2, 4, 12, 8, 40, 72, 360, 16, 112, 400, 2800, 432, 3024, 10800, 75600, 32, 352, 1568, 17248, 4000, 44000, 196000, 2156000, 2592, 28512, 127008, 1397088, 324000, 3564000, 15876000, 174636000, 64, 832, 7744, 100672, 21952, 285376, 2656192, 34530496, 40000, 520000

Offset: 0

Author

Michael De Vlieger and Peter Munn, Aug 29 2025

Keywords

• nonn
• tabf
• easy
• new

Comments

For n >= 1, row n consists of the even unitary divisors of A006939(n).

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).

For even n > 0, a(n) is powerful.

For odd n > 1, a(n) is in A332785.

a(1) = 2 is the only prime term.

a(0) = 1 and a(1) = 2 are the only squarefree terms.

a(2^k) = 2^(k+1).

Perfect powers in this sequence include A000079, but also numbers like 400 = prime(1)^4 * prime(3)^2.

Examples

			Table begins:
n\k  0    1    2     3    4     5      6       7
------------------------------------------------
0:   1;
1:   2;
2:   4,  12;
3:   8,  40,  72,  360;
4:  16, 112, 400, 2800, 432, 3024, 10800, 75600;
     ...
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:
 n     a(n)  A067255(a(n))
--------------------------
 0       1   .
 1       2   1
 2       4   2
 3      12   21
 4       8   3
 5      40   3.1
 6      72   32
 7     360   321
 8      16   4
 9     112   4..1
10     400   4.2
11    2800   4.21
12     432   43
13    3024   43.1
14   10800   432
15   75600   4321
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16384
• Michael De Vlieger, Log log scatterplot of a(n) n = 0..2^14.
• Michael De Vlieger, Plot prime(i)^j at (x,y) = (n,i), 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.

Crossrefs

Cf. A001597, A001694, A110765 (squarefree kernel), A286708, A362227, A363250, A384003.

All terms are in A304686.

See the comments for the relationships with A000079, A006939, A332785.

See the formula section for the relationships with A000120, A001221, A001222, A007947, A019565, A029837, A029931, A036044, A064549, A093141, A167747, A242378, A265127.

See the examples for the relationship with A067255.

Programs

Mathematica

f[x_] := If[x == 1, {0},
 Function[g,ReplacePart[Table[0, {PrimePi[f[[-1, 1]] ]}], #] &@
     Map[PrimePi@ First@ # -> Last@ # &, g] ]@ FactorInteger@ x];
Table[f[Reverse@ Range[Length[#]]*#] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 120}]

Formula

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.

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.

a(0) = 1; otherwise:

a(2n) = A064549(a(n)) = a(n) * rad(a(n));

a(2n+1) = a(2n) * prime(row_index(n)).

T(n,0) = 2^n.

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".

T(n,ceiling(2^(n-1))-1) = A006939(n).

For n > 1, T(n,1) = A265127(n) = 2^n * prime(n).

For n > 2, T(n,2^(n-1)) = A167747(n) = phi(6^n) = 2^n * 3^(n-1).

For n > 2, T(n,2^(n-2)) = A093141(n-1) = 4 * 10^(n-1). = 2^n * 5^(n-2).

T(n,k) = max({j >= 1 : j|A006939(n) and gcd(j, A019565(A036044(lin_index(n,k)))) = 1}).

A110765(n) = A007947(a(n)).

A001221(a(n)) = A000120(n).

A001222(a(n)) = A029931(n).