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%I #44 Jul 25 2017 02:41:38
%S 3,5,7,9,17,19,21,43,45,111,193,195,477,927,1777,1779,2973,4963,6397,
%T 6399,12961,14983,14985,40191,66993,114063,193233,334959,558273,
%U 951999,1586673,3724815,8255985,18271887,31279473,66853647,171456753,339654927
%N Natural growth of an aliquot sequence driven by a perfect number 2^(p-1)*((2^p) - 1).
%C 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).
%C 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
%H Mathworld, <a href="http://mathworld.wolfram.com/AliquotSequence.html">Aliquot Sequence</a>
%H Stern, <a href="http://www.lafn.org/~ax810/aliquot.htm">Aliquot Sequences from the trenches</a> [broken link?]
%F a(n) = a(n-1) + 2*(sigma(a(n-1)) - a(n-1)). - _Roderick MacPhee_, Aug 21 2012
%e 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.
%p 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
%t NestList[2*DivisorSigma[1,#]-#&,3,40] (* _Harvey P. Dale_, Jul 16 2013 *)
%o (PARI) A146556()=a=[3];until(#a==79,a=concat(a,a[#a]+2*(sigma(a[#a])-a[#a])));a
%o (PARI) a(n)=if(n==1,3,2*sigma(a(n-1))-a(n-1)) \\ _R. K. Guy_, Jul 16 2013
%Y Cf. A000396, A008892, A215778, A216224.
%K hard,nonn
%O 1,1
%A _Sergio Pimentel_, Oct 31 2008
%E More terms, as derived from p=7, driver 8128. - _R. J. Mathar_, Jan 22 2009
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