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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.

%I A384003 #23 Aug 30 2025 10:11:36
%S A384003 1,2,3,12,5,40,45,360,7,112,189,3024,175,2800,4725,75600,11,352,891,
%T A384003 28512,1375,44000,111375,3564000,539,17248,43659,1397088,67375,
%U A384003 2156000,5457375,174636000,13,832,3159,202176,8125,520000,1974375,126360000,4459,285376,1083537
%N 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.
%C A384003 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).
%H A384003 Michael De Vlieger, <a href="/A384003/b384003.txt">Table of n, a(n) for n = 0..16384</a> (rows n = 0..14, flattened)
%H A384003 Michael De Vlieger, <a href="/A384003/a384003.png">Log log scatterplot of a(n)</a>, n = 0..16384.
%H A384003 Michael De Vlieger, <a href="/A384003/a384003_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 = 14 in magenta.
%F A384003 T(0,0) = 1; T(1,0) = 2.
%F A384003 Otherwise, T(n,2k) = A003961(T(n-1,k)).
%F A384003 T(n,2k+1) = T(n,2k)*2^n.
%F A384003 T(n,0) = prime(n).
%F A384003 T(n,2^(n-1)-1) = A006939(n).
%F A384003 T(n,2^(n-2)) = A251720(n).
%F A384003 Using a(m) to denote a term of the linear sequence with offset 0: (Start)
%F A384003 A019565(m) = A007947(a(m)).
%F A384003 a(m) = T(n,k) = gcd(A019565(m)^n, A006939(n)).
%F A384003 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)).
%F A384003 (End)
%F A384003 For n > 0, A071178(T(n,k)) = 1.
%e A384003 Table begins:
%e A384003 n\k  0    1    2     3    4     5     6       7
%e A384003 -----------------------------------------------
%e A384003 0:   1;
%e A384003 1:   2;
%e A384003 2:   3,  12;
%e A384003 3:   5,  40,  45,  360;
%e A384003 4:   7, 112, 189, 3024, 175, 2800, 4725, 75600;
%e A384003      ...
%e A384003 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 A384003  n     a(n)  A067255(a(n))
%e A384003 --------------------------
%e A384003  0       1   .
%e A384003  1       2   1
%e A384003  2       3   .1
%e A384003  3      12   21
%e A384003  4       5   ..1
%e A384003  5      40   3.1
%e A384003  6      45   .21
%e A384003  7     360   321
%e A384003  8       7   ...1
%e A384003  9     112   4..1
%e A384003 10     189   .3.1
%e A384003 11    3024   43.1
%e A384003 12     175   ..21
%e A384003 13    2800   4.21
%e A384003 14    4725   .321
%e A384003 15   75600   4321
%t A384003 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}]
%Y A384003 Cf. A000040, A003961, A006939, A007947, A019565, A060175, A061395, A071178, A251720.
%K A384003 nonn,tabf,easy,base,changed
%O A384003 0,2
%A A384003 _Michael De Vlieger_ and _Peter Munn_, May 28 2025
%E A384003 Name edited by _Peter Munn_, Aug 30 2025