A189394 Highly composite numbers whose number of divisors is also highly composite.

1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200

Offset: 1

Author

Krzysztof Ostrowski, Apr 21 2011

Keywords

• nonn
• fini
• full

Comments

Both n and d(n) are highly composite numbers.

It is extremely likely that this sequence is complete. The highly composite numbers have a very special form. The number of divisors of a large HCN has a high power of 2 in its factorization -- which is not the form of an HCN. - T. D. Noe, Apr 21 2011

All but a(7) and a(12) are a multiple of the previous term: ratios a(n+1) / a(n) are (2, 3, 2, 5, 6, 7/2, 2, 2, 11, 5, 13/5, 5, 17, 36, 133, 23, 29, ...?). - M. F. Hasler, Jun 20 2022

Examples

			d(60) = 12; both 60 and 12 are highly composite numbers
		

Links

• Achim Flammenkamp, Highly composite numbers
• Lars Magnus Øverlier, Highly Composite Numbers, arXiv:2305.14350 [math.NT], 2023.

Crossrefs

Cf. A002182, A002183, A141320.

Programs

Mathematica

(* First run program at A002182 *) Select[A002182, MemberQ[A002182, DivisorSigma[0, #]] &] (* Alonso del Arte, Apr 21 2011 *)

PARI

is_A189394(n)={is_A002182(numdiv(n)) && is_A002182(n)}
M189394=[1,2]/*for memoization*/; A189394(n)={if(#M189394A189394(k++*s),); M189394=concat(M189394,k*s)); M189394[n]} \\ M. F. Hasler, Jun 20 2022

Extensions

Typo in a(15) corrected by Ben Beer, Jul 20 2016

Keywords fini and full, following Øverlier's thesis, added by Michel Marcus, May 25 2023