A384003 Irregular triangle T(n,k), n >= 0, 0 <= k < 2^(n-1), where T(n,k) = Product_{j=0..n-1} prime(j+1)^((n-j)*d_j), where d_j is the bit with digit weight 2^j in the binary expansion of 2^(n-1)+k.

1, 2, 3, 12, 5, 40, 45, 360, 7, 112, 189, 3024, 175, 2800, 4725, 75600, 11, 352, 891, 28512, 1375, 44000, 111375, 3564000, 539, 17248, 43659, 1397088, 67375, 2156000, 5457375, 174636000, 13, 832, 3159, 202176, 8125, 520000, 1974375, 126360000, 4459, 285376, 1083537

Offset: 0

Author

Michael De Vlieger and Peter Munn, May 28 2025

Keywords

• nonn
• tabf
• easy
• base
• changed

Comments

This sequence can be seen as a structured ordering of numbers m that are not divisible by the square of their greatest prime factor and where every prime in the canonical factorization of m has the same sum of prime index and exponent. For example, prime(1)^3 * prime(3)^1 = 2^3 * 5 = 40. The ordering is lexicographic according to prime divisors listed in decreasing order, as used for A019565. Row n has the numbers whose greatest prime factor is prime(n).

Examples

			Table begins:
n\k  0    1    2     3    4     5     6       7
-----------------------------------------------
0:   1;
1:   2;
2:   3,  12;
3:   5,  40,  45,  360;
4:   7, 112, 189, 3024, 175, 2800, 4725, 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       3   .1
 3      12   21
 4       5   ..1
 5      40   3.1
 6      45   .21
 7     360   321
 8       7   ...1
 9     112   4..1
10     189   .3.1
11    3024   43.1
12     175   ..21
13    2800   4.21
14    4725   .321
15   75600   4321
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16384 (rows n = 0..14, flattened)
• Michael De Vlieger, Log log scatterplot of a(n), n = 0..16384.
• 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 = 14 in magenta.

Crossrefs

Cf. A000040, A003961, A006939, A007947, A019565, A060175, A061395, A071178, A251720.

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

T(0,0) = 1; T(1,0) = 2.

Otherwise, T(n,2k) = A003961(T(n-1,k)).

T(n,2k+1) = T(n,2k)*2^n.

T(n,0) = prime(n).

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

T(n,2^(n-2)) = A251720(n).

Using a(m) to denote a term of the linear sequence with offset 0: (Start)

A019565(m) = A007947(a(m)).

a(m) = T(n,k) = gcd(A019565(m)^n, A006939(n)).

Equivalently, for p = A000040(i), the i-th prime, p|a(m) iff p|A019565(m), in which case A060175(m,i) = j - i + 1, where j = PrimePi(gpf(A019565(m))) = A061395(A019565(m)).

(End)

For n > 0, A071178(T(n,k)) = 1.

Extensions

Name edited by Peter Munn, Aug 30 2025