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

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

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 b-file)
/* 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, m-n>r, print1([i, n, r=m-n]", ")))
\\ M. F. Hasler, Jan 08 2020

Crossrefs

Cf. A002182, A002183, A108602, A112779, A112780, A112781.

Sequence in context: A071475 A358104 A343228 * A080594 A207291 A364443

Adjacent sequences: A112775 A112776 A112777 * A112779 A112780 A112781

Keyword

nonn,changed

Author

Ray Chandler, Nov 11 2005

Status

approved