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A347042
Number of divisors d > 1 of n such that bigomega(d) divides bigomega(n), where bigomega = A001222.
6
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
1,4
LINKS
EXAMPLE
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
MATHEMATICA
Table[Length[Select[Rest[Divisors[n]], IntegerQ[PrimeOmega[n]/PrimeOmega[#]]&]], {n, 100}]
PROG
(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
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.
Sequence in context: A154263 A293435 A294901 * A333416 A305818 A303757
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 17 2021
STATUS
approved