A377070 - OEIS

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.

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.

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.

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

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}]