A084931 Consider trajectory of n under repeated applications of the function f(x) = 'Sum of the prime factors of x (with multiplicity)' (see A029908). Sequence gives composite numbers n that end at a prime m that divides n and m is greater than any m's seen already.

15, 21, 182, 494, 1219, 2852, 3182, 9782, 19339, 19982, 22454, 72836, 76814, 102134, 156782, 192182, 423182, 750979, 758894, 1364534, 1465454, 1548782, 2376182, 3379982, 4066934, 4204982

Offset: 1

Author

Sven Simon, Jun 12 2003

Keywords

• easy
• nonn

Comments

With a prime triple (p,p+4,p+6), the number a(n) = 2*p*(p+6) is always in the sequence, f( f( 2*p*(p+6) )) = f( 2*(p+4) ) = p+6. Such prime triples can be found in sequence A022005.

As long as two successive triples (p1,p1 + 4,p1 + 6) and (p2,p2 + 4,p2 + 6) of A022005 have p2 < 1.2*p1, no other numbers occur in the sequence between a(n1) and a(n2), this holds at least for larger p1 > 500. Other types of prime sets occurring in the sequence: (p,p+4,3p-4) with F( F( (p+4)*(3p-4))) = F( 4p ) = p + 4 (p,p+6,p+8) with F( F( 4*p*(p+8) )) = F( 2*(p+6) ) = p + 8.

Large examples of (p,p+4,++6)-triples: (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, + 13 (4135 digits, David Broadhurst) (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, +13 (4134 digits, David Broadhurst) Record examples of prime triples can be found on Tony Forbes's web site. There are triples of type (p,p+4,p+6) too.

Examples

			a(10) = 19982: f(f(19982)) = f(f(2*97*103)) = f(2+97+103) = f(202) = f(2*101) = 2+101 = 103.
		

Crossrefs

Cf. A022005, A048133, A084932 (primes reached).