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%I #32 May 25 2023 07:08:37
%S 1,2,6,12,60,360,1260,2520,5040,55440,277200,720720,3603600,61261200,
%T 2205403200,293318625600,6746328388800,195643523275200
%N Highly composite numbers whose number of divisors is also highly composite.
%C Both n and d(n) are highly composite numbers.
%C 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
%C 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
%H Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">Highly composite numbers</a>
%H Lars Magnus Øverlier, <a href="https://arxiv.org/abs/2305.14350">Highly Composite Numbers</a>, arXiv:2305.14350 [math.NT], 2023.
%e d(60) = 12; both 60 and 12 are highly composite numbers
%t (* First run program at A002182 *) Select[A002182, MemberQ[A002182, DivisorSigma[0, #]] &] (* _Alonso del Arte_, Apr 21 2011 *)
%o (PARI) is_A189394(n)={is_A002182(numdiv(n)) && is_A002182(n)}
%o 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
%Y Cf. A002182, A002183, A141320.
%K nonn,fini,full
%O 1,2
%A _Krzysztof Ostrowski_, Apr 21 2011
%E Typo in a(15) corrected by _Ben Beer_, Jul 20 2016
%E Keywords fini and full, following Øverlier's thesis, added by _Michel Marcus_, May 25 2023
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