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

0,2

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.

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.

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).

Example

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

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}]

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.

Sequence in context: A363250 A181817 A362227 * A381931 A386822 A000348

Adjacent sequences: A387462 A387463 A387464 * A387466 A387467 A387468

Keyword

nonn,tabf,easy

Author

Michael De Vlieger and Peter Munn, Aug 29 2025

Status

approved