A307111 a(n) is the least primitive n-abundant number k with the largest possible abundancy index sigma(k)/k.

3465, 6930, 19399380, 8172244080

Offset: 2

Author

Amiram Eldar, Mar 25 2019

Keywords

• nonn
• hard
• bref
• more

Comments

A primitive n-abundant number k is a number with sigma(k)/k > n and sigma(d)/d < n for all the proper divisors d of k.

Cohen proved that for any given eps > 0 there are only finitely many primitive n-abundant numbers k with sigma(k)/k >= n + eps. Thus for each n there is a maximal value of the abundancy index sigma(k)/k for primitive n-abundant numbers k. If this maximum occurs at more than one number the least of them is given in this sequence.

Examples

			3465 is in the sequence since it is the primitive abundant (A071395) number with the largest possible abundancy index among the primitive abundant numbers: sigma(3465)/3465 = 832/385 = 2.161003... The abundancy indices of the next terms are 1248/385 = 3.241558..., 193536/46189 = 4.190088..., 642816/124729 = 5.153701...
		

Links

• Graeme L. Cohen, Primitive alpha-abundant numbers, Mathematics of Computation, Vol. 43, No. 167 (1984), pp. 263-270.

Crossrefs

Cf. A000203, A071395, A307112, A307114, A307115.