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

1,2

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

Links

Table of n, a(n) for n=1..18.

Achim Flammenkamp, Highly composite numbers

Lars Magnus Øverlier, Highly Composite Numbers, arXiv:2305.14350 [math.NT], 2023.

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A002182, A002183, A141320.

Sequence in context: A309875 A254232 A335831 * A182862 A072938 A160274

Adjacent sequences: A189391 A189392 A189393 * A189395 A189396 A189397

Keyword

nonn,fini,full

Author

Krzysztof Ostrowski, Apr 21 2011

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

Status

approved