A260588 -id:A260588 - cp's OEIS frontend

A260587 Number of distinct prime factors of A173426(n) = concatenation of (1, 2, ..., n, n-1, ..., 1).

0, 1, 2, 2, 2, 5, 2, 4, 3, 1, 2, 3, 3, 6, 6, 4, 4, 4, 6, 2, 5, 3, 4, 8, 2, 6, 8, 2, 4, 9, 4, 9, 6, 6, 6, 7, 3, 5, 7, 4, 6, 6, 3, 6

Offset: 1

M. F. Hasler, Jul 29 2015

			a(2) = 1 since A173426(2) = 121 = 11*11 has only one distinct prime factor, 11.
a(21) = 5 since A173426(21) = 3^2 * 7 * 828703 * 94364768151913037621 * 250591098443370396365457961250972909.
a(25) = 2 since A173426(25) = A075023(n) * A075024(n) is a semiprime.

FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.

Cf. A001221.
See A260588 for the variant where prime factors are counted with multiplicity.
See also A075023 and A075024 for the smallest and largest prime factor of the terms.

PARI
```
a(n)=omega(A173426(n))
```

a(38)-a(44) added using factordb.com by Jinyuan Wang, Mar 05 2020

A260589 Irregular table read by rows: n-th row lists the prime factors of A173426(n), with repetition.

Original entry on oeis.org

11, 11, 3, 3, 37, 37, 11, 11, 101, 101, 41, 41, 271, 271, 3, 3, 7, 7, 11, 11, 13, 13, 37, 37, 239, 239, 4649, 4649, 11, 11, 73, 73, 101, 101, 137, 137, 3, 3, 3, 3, 37, 37, 333667, 333667, 12345678910987654321, 7, 17636684157301569664903, 3, 3, 7, 7, 2799473675762179389994681, 1109, 4729

Offset: 1

M. F. Hasler, Jul 29 2015

Row lengths are given by A260588(n). In particular, row n = 1 would have length 0, i.e., no element, because A173426(1) = 1 has no prime factors. Therefore the sequence can be considered to start with row n = 2. (The offset refers to the k-th element of the "flattened" sequence.)

For n = 1 through n = 9, A173426(n) is the square of the repunit 1...1 of length n, therefore every prime factor appears twice. This is no longer the case for n > 9.

M. F. Hasler, Table of k, a(k) for k = 1..150 (Rows n = 2 through 30, flattened.)
M. F. Hasler, Factorization of A173426 = 123...321, OEIS wiki, July 2015.

Cf. A001222, A075023, A075024, A173426, A260587, A260588.

PARI
```
A260589_row(n)=A027746_row(A173426(n))
```

vector(30,n,A027746_row(A173426(n))) \\ You may concat() this.

  n |  A173426(n) | factors = n-th row of this table
|      1      | []
|     121     | [11, 11]
|    12321    | [3, 3, 37, 37]
|   1234321   | [11, 11, 101, 101]
|  123454321  | [41, 41, 271, 271]
| 12345654321 | [3, 3, 7, 7, 11, 11, 13, 13, 37, 37]

cp's OEIS Frontend

A260587 Number of distinct prime factors of A173426(n) = concatenation of (1, 2, ..., n, n-1, ..., 1).

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A260589 Irregular table read by rows: n-th row lists the prime factors of A173426(n), with repetition.

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