A347042 - OEIS

%I #23 Nov 02 2024 09:14:17

%S 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,

%T 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,

%U 3,4,1,3,1,3,3,3,3,4,1,3,3,3,1,8,3,3,3

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

%H Amiram Eldar, <a href="/A347042/b347042.txt">Table of n, a(n) for n = 1..10000</a>

%e The a(n) divisors for selected n:

%e   n = 1:   2:   4:   6:   24:  30:  36:  60:  96:  144: 210: 216: 240: 360:

%e       ---------------------------------------------------------------------

%e       {}   2    2    2    2    2    2    2    2    2    2    2    2    2

%e                 4    3    3    3    3    3    3    3    3    3    3    3

%e                      6    4    5    4    4    4    4    5    4    4    4

%e                           6    30   6    5    6    6    6    6    5    5

%e                           24        9    6    8    8    7    8    6    6

%e                                     36   10   12   9    10   9    8    8

%e                                          15   96   12   14   12   10   9

%e                                          60        18   15   18   12   10

%e                                                    144  21   27   15   12

%e                                                         35   216  20   15

%e                                                         210       30   18

%e                                                                   240  20

%e                                                                        30

%e                                                                        45

%e                                                                        360

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

%o (PARI) a(n) = my(bn=bigomega(n)); sumdiv(n, d, if (d>1, !(bn % bigomega(d)))); \\ _Michel Marcus_, Aug 18 2021

%o (Python)

%o from sympy import divisors, primeomega

%o def a(n):

%o     bigomegan = primeomega(n)

%o     return sum(bigomegan%primeomega(d) == 0 for d in divisors(n)[1:])

%o print([a(n) for n in range(1, 88)]) # _Michael S. Branicky_, Aug 18 2021

%o (Python)

%o from sympy import factorint, divisors

%o from sympy.utilities.iterables import multiset_combinations

%o def A347042(n):

%o     fs = factorint(n,multiple=True)

%o     return sum(len(list(multiset_combinations(fs,d))) for d in divisors(len(fs),generator=True)) # _Chai Wah Wu_, Aug 21 2021

%Y Positions of 1's are A000040.

%Y The smallest of these divisors is A020639

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

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors.

%Y A001222 counts all prime factors, also called bigomega.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A207375 lists central divisors.

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 17 2021