A046952 Sets record for f(n) = |{(a,b):a*b=n and a|b}|. Also squares of highly composite numbers A002182.

1, 4, 16, 36, 144, 576, 1296, 2304, 3600, 14400, 32400, 57600, 129600, 518400, 705600, 1587600, 2822400, 6350400, 25401600, 57153600, 101606400, 228614400, 406425600, 635040000, 768398400, 2057529600, 2540160000, 3073593600

Offset: 1

Author

Simon Colton (simonco(AT)cs.york.ac.uk)

Keywords

• nice
• nonn

Comments

Invented by the HR automatic theory formation program.

Also, integers whose number of square divisors sets a new record. - Bernard Schott, Jan 14 2022

As a(n) is the square of n-th highly composite number (A002182), the record number of square divisors of a(n) is A046951(a(n)) = tau(A002182(n)) = A002183(n) where tau is the divisor counting function (A000005). - Bernard Schott, Jan 15 2022

Integers m for which number of solutions (A353282) to the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = m sets a new record; these records are respectively 0, 1, 2, 3, 5, 7, ... Example: the 5 solutions for S(x,y) = 144 are (36,1), (32,2), (27,3), (20,5), (11,11). - Bernard Schott, Apr 19 2022

Examples

			f(1)=1, (first with 1), f(4)=2 (first with 2), f(16)=3 (first with 3).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
• Simon Colton, HR - Automatic Theory Formation in Pure Mathematics

Crossrefs

Cf. A000005, A002182, A002183, A046951.

Cf. A350756 (similar, with triangular divisors).

Formula

a(n) = A002182(n)^2. - Bernard Schott, Jan 14 2022