A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

1, 1, 1, 2, 1, 3, 1, 1, 3, 5, 1, 1, 1, 7, 5, 4, 1, 1, 1, 1, 7, 11, 1, 6, 5, 13, 1, 1, 1, 1, 1, 1, 11, 17, 7, 9, 1, 19, 13, 10, 1, 1, 1, 1, 1, 23, 1, 1, 7, 1, 17, 1, 1, 9, 11, 14, 19, 29, 1, 15, 1, 31, 1, 8, 13, 1, 1, 1, 23, 1, 1, 1, 1, 37, 1, 1, 11, 1, 1, 1, 9

Offset

1,4

Comments

Problem: What are the positions of last appearances > 1?

Links

Table of n, a(n) for n=1..81.

Example

The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 15.

Mathematica

Table[If[#=={}, 1, Max[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&], {n, 100}]

Prog

(Python)
from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[-1:0:-1]:
        if 2*len(factorint(d, multiple=True)) == npf: return d
    return 1
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Aug 18 2021
(Python 3.8+)
from math import prod
from sympy import factorint
def A347046(n):
    fs = factorint(n, multiple=True)
    q, r = divmod(len(fs), 2)
    return 1 if r else prod(fs[q:]) # Chai Wah Wu, Aug 20 2021

Crossrefs

The greatest divisor without the condition is A006530 (smallest: A020639).

Positions of 1's are A026424.

The case of powers of 2 is A072345.

Positions of first appearances are A123667 (sorted: A123666).

Divisors of this type are counted by A345957 (rounded: A096825).

The rounded version is A347044.

The smallest divisor of this is A347045 (rounded: A347043).

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 (min: A033676, max: A033677).

A340387 lists numbers whose sum of prime indices is twice bigomega.

A340609 lists numbers whose maximum prime index divides bigomega.

A340610 lists numbers whose maximum prime index is divisible by bigomega.

A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).

Cf. A000720, A038548, A060775, A106529, A244991, A335433, A335448.

Sequence in context: A367412 A279945 A342724 * A300322 A144220 A156826

Adjacent sequences: A347043 A347044 A347045 * A347047 A347048 A347049

Keyword

nonn

Author

Gus Wiseman, Aug 16 2021

Status

approved