

A112778


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


14



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, 11, 10, 11, 11, 11, 12, 10, 10, 11, 11, 12, 11, 12, 12, 12, 13, 12, 13, 14, 13, 13, 12, 14, 12, 13, 13, 13, 14, 13, 14, 15, 14, 14, 13, 15, 15, 14, 14, 16, 14, 15, 14, 15, 16, 15, 15, 16
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OFFSET

1,3


COMMENTS

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


LINKS

Joerg Arndt, Table of n, a(n) for n = 1..19999
Eric Weisstein's World of Mathematics, Highly Composite Number


FORMULA

a(n) = A001222(A002182(n)).


EXAMPLE

A002182(8) = 48 = 2^4*3, which has 5 prime factors, counted with multiplicity, so a(8)=5.


PROG

(PARI)
A112778(n)=bigomega(A002182(n)) \\ or A112778(n)=v112778[n] (e.g., from bfile)
/* To list the records of max(a(1..n))  a(n): */
m=r=0; for(i=1, 1e4, if(m<n=A112778(i), m=n, mn>r, print1([i, n, r=mn]", ")))
\\ M. F. Hasler, Jan 08 2020


CROSSREFS

Cf. A002182, A002183, A108602, A112779, A112780, A112781.
Sequence in context: A141525 A209764 A071475 * A080594 A207291 A194175
Adjacent sequences: A112775 A112776 A112777 * A112779 A112780 A112781


KEYWORD

nonn


AUTHOR

Ray Chandler, Nov 11 2005


STATUS

approved



