Probably pure coincidence but if you look at the deconstruction of the dedekinds like so:

>>> decon(6)
offset: 1, exp: [[Decimal('2'), Decimal('1')], [Decimal('3'), Decimal('1')]]

>>> decon(20)
offset: 2, exp: [[Decimal('2'), Decimal('2')], [Decimal('5'), Decimal('1')]]
offset: 1, exp: []

>>> decon(168)
offset: 3, exp: [[Decimal('2'), Decimal('2')], [Decimal('5'), Decimal('2')]]
offset: 2, exp: [[Decimal('2'), Decimal('2')], [Decimal('3'), Decimal('1')], [Decimal('5'), Decimal('1')]]
offset: 1, exp: [[Decimal('2'), Decimal('3')]]

>>> decon(7581)
offset: 4, exp: [[Decimal('2'), Decimal('3')], [Decimal('5'), Decimal('3')], [Decimal('7'), Decimal('1')]]
offset: 3, exp: [[Decimal('2'), Decimal('2')], [Decimal('5'), Decimal('3')]]
offset: 2, exp: [[Decimal('2'), Decimal('4')], [Decimal('5'), Decimal('1')]]
offset: 1, exp: []

>>> decon(7828354)
offset: 7, exp: [[Decimal('2'), Decimal('6')], [Decimal('5'), Decimal('6')], [Decimal('7'), Decimal('1')]]
offset: 6, exp: [[Decimal('2'), Decimal('8')], [Decimal('5'), Decimal('5')]]
offset: 5, exp: [[Decimal('2'), Decimal('5')], [Decimal('5'), Decimal('4')]]
offset: 4, exp: [[Decimal('2'), Decimal('6')], [Decimal('5'), Decimal('3')]]
offset: 3, exp: [[Decimal('2'), Decimal('2')], [Decimal('3'), Decimal('1')], [Decimal('5'), Decimal('2')]]
offset: 2, exp: [[Decimal('2'), Decimal('1')], [Decimal('5'), Decimal('2')]]
offset: 1, exp: [[Decimal('2'), Decimal('2')]]

>>> decon(d('2414682040998'))
offset: 13, exp: [[Decimal('2'), Decimal('13')], [Decimal('5'), Decimal('12')]]
offset: 12, exp: [[Decimal('2'), Decimal('13')], [Decimal('5'), Decimal('11')]]
offset: 11, exp: [[Decimal('2'), Decimal('10')], [Decimal('5'), Decimal('10')]]
offset: 10, exp: [[Decimal('2'), Decimal('11')], [Decimal('5'), Decimal('9')]]
offset: 9, exp: [[Decimal('2'), Decimal('9')], [Decimal('3'), Decimal('1')], [Decimal('5'), Decimal('8')]]
offset: 8, exp: [[Decimal('2'), Decimal('10')], [Decimal('5'), Decimal('7')]]
offset: 7, exp: [[Decimal('2'), Decimal('7')], [Decimal('5'), Decimal('6')]]
offset: 6, exp: []
offset: 5, exp: [[Decimal('2'), Decimal('6')], [Decimal('5'), Decimal('4')]]
offset: 4, exp: []
offset: 3, exp: [[Decimal('2'), Decimal('2')], [Decimal('3'), Decimal('2')], [Decimal('5'), Decimal('2')]]
offset: 2, exp: [[Decimal('2'), Decimal('1')], [Decimal('3'), Decimal('2')], [Decimal('5'), Decimal('1')]]
offset: 1, exp: [[Decimal('2'), Decimal('3')]]

the powers in the 2's column go:
1, 2, 2, 2, 3, 3, 2, 4, 6

which are predicted by:
https://oeis.org/search/...

Again, probably only a coincidence, but kinda beautiful.