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A112778 Number of prime factors (counted with multiplicity) of highly composite numbers (definition 1, A002182).

%I A112778 #14 Feb 16 2025 08:32:59
%S A112778 0,1,2,2,3,4,4,5,4,5,5,6,6,7,6,6,7,7,8,8,9,9,10,9,8,10,10,9,9,10,10,
%T A112778 11,10,11,11,11,12,10,10,11,11,12,11,12,12,12,13,12,13,14,13,13,12,14,
%U A112778 12,13,13,13,14,13,14,15,14,14,13,15,15,14,14,16,14,15,14,15,16,15,15,16
%N A112778 Number of prime factors (counted with multiplicity) of highly composite numbers (definition 1, A002182).
%C A112778 The values of this sequence oscillate around a slowly increasing moving average, with an amplitude roughly equal to log(a(n)): Records 1, 2, 3, ... of max(a(1..n)) - a(n) are reached at n = (9, 25, 11, 307, 1201, 7140, ...) where a(n) = (4, 8, 18, 31, 64, 169, 175, ...). - _M. F. Hasler_, Jan 08 2020
%H A112778 Joerg Arndt, <a href="/A112778/b112778.txt">Table of n, a(n) for n = 1..19999</a>
%H A112778 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly Composite Number</a>
%F A112778 a(n) = A001222(A002182(n)).
%e A112778 A002182(8) = 48 = 2^4*3, which has 5 prime factors, counted with multiplicity, so a(8)=5.
%o A112778 (PARI)
%o A112778 A112778(n)=bigomega(A002182(n)) \\ or A112778(n)=v112778[n] (e.g., from b-file)
%o A112778 /* To list the records of max(a(1..n)) - a(n): */
%o A112778 m=r=0; for(i=1,1e4, if(m<n=A112778(i), m=n, m-n>r, print1([i,n,r=m-n]",")))
%o A112778 \\ _M. F. Hasler_, Jan 08 2020
%Y A112778 Cf. A002182, A002183, A108602, A112779, A112780, A112781.
%K A112778 nonn
%O A112778 1,3
%A A112778 _Ray Chandler_, Nov 11 2005