A146556 Natural growth of an aliquot sequence driven by a perfect number 2^(p-1)*((2^p) - 1).

3, 5, 7, 9, 17, 19, 21, 43, 45, 111, 193, 195, 477, 927, 1777, 1779, 2973, 4963, 6397, 6399, 12961, 14983, 14985, 40191, 66993, 114063, 193233, 334959, 558273, 951999, 1586673, 3724815, 8255985, 18271887, 31279473, 66853647, 171456753, 339654927

Offset: 1

Author

Sergio Pimentel, Oct 31 2008

Keywords

• hard
• nonn

Comments

This is the natural growth of an aliquot sequence that has a driver of the form 2^(p-1) * ((2^p) - 1) (Perfect Number). It will continue growing this way until it loses the driver, which can only happen when the next term and the driver are not coprimes (which hardly ever happens).

The natural growth of the aliquot sequence starting with p=5 at 2^(p-1)*(2^p-1)*3 = 496*3 = 1488 has the factors 3, 5, 7, 9, 17, 19, 21, 43, 45, 111, 193, 195, 477, 927, 1777, 1779, 2973, 4963, 6397, 6399, 12961, 14983, 14985, 40191, 66993, 114063, 193233, 334959, 558273, 951999, 1586673, 3564018 and "loses the driver" at the next term because it is not a multiple of 496. I complemented the terms therefore from p=7 and initial factor 3 which does not lose the driver early. - R. J. Mathar, Jan 22 2009

Examples

			The aliquot sequence starting at 1488 (2^4*31*3) is: 1488, 2480, 3472, 4464,8432, 9424 or: 496*3, 496*5, 496*7, 496*9, 496*17, 496*19, always keeping the 496 driver until reaching a term that is not coprime with 496.
		

Links

• Mathworld, Aliquot Sequence
• Stern, Aliquot Sequences from the trenches [broken link?]

Crossrefs

Cf. A000396, A008892, A215778, A216224.

Programs

Maple

p := 7: dr := 2^(p-1)*(2^p-1) ; f := 3 ; aliq := proc(n) option remember ; global dr,f ; local an_1 ; if n = 1 then dr*f ; else an_1 := procname(n-1) ; numtheory[sigma](an_1)-an_1 ; fi; end: A := proc(n) option remember ; global dr ; aliq(n)/dr ; end: for n from 1 to 70 do printf("%a,",A(n)) ; od: # R. J. Mathar, Jan 22 2009

Mathematica

NestList[2*DivisorSigma[1,#]-#&,3,40] (* Harvey P. Dale, Jul 16 2013 *)

PARI

A146556()=a=[3];until(#a==79,a=concat(a,a[#a]+2*(sigma(a[#a])-a[#a])));a

PARI

a(n)=if(n==1,3,2*sigma(a(n-1))-a(n-1)) \\ R. K. Guy, Jul 16 2013

Formula

a(n) = a(n-1) + 2*(sigma(a(n-1)) - a(n-1)). - Roderick MacPhee, Aug 21 2012

Extensions

More terms, as derived from p=7, driver 8128. - R. J. Mathar, Jan 22 2009