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

Author

Ray Chandler, Nov 11 2005

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

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

Programs

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(mA112778(i), m=n, m-n>r, print1([i,n,r=m-n]",")))
\\ M. F. Hasler, Jan 08 2020

Formula

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