A377070 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) = bigomega(n), where rad = A007947 and bigomega = A001222.

1, 2, 3, 4, 5, 4, 6, 9, 7, 8, 9, 4, 10, 25, 11, 8, 12, 18, 27, 13, 4, 14, 49, 9, 15, 25, 16, 17, 8, 12, 18, 27, 19, 8, 20, 50, 125, 9, 21, 49, 4, 22, 121, 23, 16, 24, 36, 54, 81, 25, 4, 26, 169, 27, 8, 28, 98, 343, 29, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 31

Offset: 1

Author

Michael De Vlieger, Oct 25 2024

Keywords

• nonn
• easy
• tabf

Comments

Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k = bigomega(n), that is, numbers m such that rad(m) | n and m has the same number of prime factors with repetition than does n.

Examples

			Triangle begins:
    n    row n of this sequence:
   -------------------------------------------
    1:   {1}
    2:   {2}
    3:   {3}
    4:   {4}
    5:   {5}
    6:   {4, 6, 9}
    7:   {7}
    8:   {8}
    9:   {9}
   10:   {4, 10, 25}
   ...                       (Select rows appear below)
   12:   {8, 12, 18, 27}
   14:   {4, 14, 49}
   15:   {9, 15, 25}
   18:   {8, 12, 18, 27}
   20:   {8, 20, 50, 125}
   24:   {16, 24, 36, 54, 81}
   30:   {8, 12, 18, 20, 27, 30, 45, 50, 75, 125}
   42:   {8, 12, 18, 27, 28, 42, 63, 98, 147, 343}
   60:   {16, 24, 36, 40, 54, 60, 81, 90, 100, 135, 150, 225, 250, 375, 625}.
.
Diagrams of the rank omega(n)-1 simplexes created by row n of this sequence for select n, ordering k in row n by prime decomposition. The number k = n appears in brackets:
Rank 3:
   n = 30:                    n = 42:
             8                         8
           /  \                      /  \
         12 -- 20                  12 -- 28
        /  \  /  \                /  \  /  \
      18 --[30]-- 50            18 --[42]-- 98
     /  \  /  \  /  \          /  \  /  \  /  \
   27 -- 45 -- 75 -- 125     27 -- 63 --147 -- 343
.
   n = 60:     16
              /  \
            24 -- 40
           /  \  /  \
         36 --[60]-- 50
        /  \  /  \  /  \
      54 -- 90 -- 75 --125
     /  \  /  \  /  \  /  \
   81 --150 --135 --375 --625
.
Rank 4:
   n = 210:
   16
        40
   24   56
             100
        60   140
   36   84   196
                   250
             150   350
        90  [210]  490
   54  126   294   686
                            625
                     375    875
              225    525   1225
        135   315    735   1715
   81   189   441   1029   2401
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..12021, (rows n = 1..1500, flattened)
• Michael De Vlieger, Diagrams of select a(n) illustrating rank omega(n)-1 simplexes formed by the arrangement of terms in row n by prime power decomposition.
• Michael De Vlieger, Log log scatterplot of a(n), rows n = 1..65536 (1278755 terms).

Crossrefs

Cf. A001221, A001222, A007947, A024619, A376248, A377071.

Programs

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[k = PrimeOmega[n]; Select[Range[n^PrimeNu[n]], Divisible[n, rad[#]] && PrimeOmega[#] == k &], {n, 30}]

Formula

Row n of this sequence is { m : rad(m) | n, bigomega(m) = bigomega(n) }.

For prime p, row p of this sequence is {p}, generally for prime power p^k, row p^k of this sequence is {p^k}.

For n in A024619, row n of this sequence has more than 1 term.

A377071(n) = length of row n of this sequence.