A302170 -id:A302170 - cp's OEIS frontend

A304038 Irregular triangle T(n,k) read by rows: first row is 0, n-th row (n > 1) lists indices of distinct primes dividing n.

0, 1, 2, 1, 3, 1, 2, 4, 1, 2, 1, 3, 5, 1, 2, 6, 1, 4, 2, 3, 1, 7, 1, 2, 8, 1, 3, 2, 4, 1, 5, 9, 1, 2, 3, 1, 6, 2, 1, 4, 10, 1, 2, 3, 11, 1, 2, 5, 1, 7, 3, 4, 1, 2, 12, 1, 8, 2, 6, 1, 3, 13, 1, 2, 4, 14, 1, 5, 2, 3, 1, 9, 15, 1, 2, 4, 1, 3, 2, 7, 1, 6, 16, 1, 2, 3, 5, 1, 4, 2, 8, 1, 10, 17, 1, 2, 3, 18, 1, 11

Offset: 1

Ilya Gutkovskiy, May 05 2018

			The irregular triangle begins:
1:  {0}
2:  {1}
3:  {2}
4:  {1}
5:  {3}
6:  {1, 2}
7:  {4}
8:  {1}
9:  {2}
10: {1, 3}
11: {5}
12: {1, 2}

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A001221 (row lengths), A027748, A055396, A061395, A066328 (row sums), A112798, A156061 (row products), A302170.

Flatten[Table[PrimePi[FactorInteger[n][[All, 1]]], {n, 1, 62}]]

T(n,k) = A000720(A027748(n,k)).
T(n,1) = A055396(n).
T(n,A001221(n)) = A061395(n).

A355079 Irregular triangle read by rows: the first row is 1, and the n-th row (n > 1) lists the factors f of n where n/f is prime (the maximal factors of n.)

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 5, 1, 4, 6, 1, 2, 7, 3, 5, 8, 1, 6, 9, 1, 4, 10, 3, 7, 2, 11, 1, 8, 12, 5, 2, 13, 9, 4, 14, 1, 6, 10, 15, 1, 16, 3, 11, 2, 17, 5, 7, 12, 18, 1, 2, 19, 3, 13, 8, 20, 1, 6, 14, 21, 1, 4, 22, 9, 15, 2, 23, 1, 16, 24, 7, 10, 25

Offset: 1

Brian Chess, Sep 17 2022

If n is prime, then 1 is its only maximal factor.

In order for a player to select a number in the game Taxman, at least one of the number's maximal factors must be available to be claimed by the taxman.

			Triangle begins:
   1:  1
   2:  1
   3:  1
   4:  2
   5:  1
   6:  2 3
   7:  1
   8:  4
   9:  3
  10:  2 5
  11:  1
  12:  4 6
  13:  1
  14:  2 7
  15:  3 5
  16:  8
  17:  1
  18:  6 9
  19:  1
  20:  4 10

Cf. A019312 (taxman sequence), A302170.

a355079 n k = a355079_tabl !! (n-1) !! (k-1)
a355079_tabl = map a355079_row [1..]
a355079_row n = [div n x | x <- a302170_row n]

Table[n / Reverse @ FactorInteger[n][[;;, 1]], {n, 1, 50}] // Flatten (* Amiram Eldar, Sep 21 2022 *)

row(n) = if (n==1, [1], select(x->isprime(n/x), divisors(n))); \\ Michel Marcus, Sep 21 2022

from sympy import factorint
def row(n): return [1] if n < 2 else sorted(n//p for p in factorint(n))
print([an for r in range(1, 51) for an in row(r)]) # Michael S. Branicky, Sep 18 2022

T(n,k) = n / A302170(n,k).

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A304038 Irregular triangle T(n,k) read by rows: first row is 0, n-th row (n > 1) lists indices of distinct primes dividing n.

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A355079 Irregular triangle read by rows: the first row is 1, and the n-th row (n > 1) lists the factors f of n where n/f is prime (the maximal factors of n.)

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