A347042 Number of divisors d > 1 of n such that bigomega(d) divides bigomega(n), where bigomega = A001222.

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

Offset: 1

Author

Gus Wiseman, Aug 17 2021

Keywords

• nonn

Examples

			The a(n) divisors for selected n:
  n = 1:   2:   4:   6:   24:  30:  36:  60:  96:  144: 210: 216: 240: 360:
      ---------------------------------------------------------------------
      {}   2    2    2    2    2    2    2    2    2    2    2    2    2
                4    3    3    3    3    3    3    3    3    3    3    3
                     6    4    5    4    4    4    4    5    4    4    4
                          6    30   6    5    6    6    6    6    5    5
                          24        9    6    8    8    7    8    6    6
                                    36   10   12   9    10   9    8    8
                                         15   96   12   14   12   10   9
                                         60        18   15   18   12   10
                                                   144  21   27   15   12
                                                        35   216  20   15
                                                        210       30   18
                                                                  240  20
                                                                       30
                                                                       45
                                                                       360
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of 1's are A000040.

The smallest of these divisors is A020639

The case of divisors with half bigomega is A345957 (rounded: A096825).

A000005 counts divisors.

A001221 counts distinct prime factors.

A001222 counts all prime factors, also called bigomega.

A056239 adds up prime indices, row sums of A112798.

A207375 lists central divisors.

Cf. A026424, A033676, A033677, A335433, A335448, A347045, A347046.

Programs

Mathematica

Table[Length[Select[Rest[Divisors[n]],IntegerQ[PrimeOmega[n]/PrimeOmega[#]]&]],{n,100}]

PARI

a(n) = my(bn=bigomega(n)); sumdiv(n, d, if (d>1, !(bn % bigomega(d)))); \\ Michel Marcus, Aug 18 2021

Python

from sympy import divisors, primeomega
def a(n):
    bigomegan = primeomega(n)
    return sum(bigomegan%primeomega(d) == 0 for d in divisors(n)[1:])
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 18 2021

Python

from sympy import factorint, divisors
from sympy.utilities.iterables import multiset_combinations
def A347042(n):
    fs = factorint(n,multiple=True)
    return sum(len(list(multiset_combinations(fs,d))) for d in divisors(len(fs),generator=True)) # Chai Wah Wu, Aug 21 2021