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

%I A146556 #45 Feb 16 2025 08:33:09
%S A146556 3,5,7,9,17,19,21,43,45,111,193,195,477,927,1777,1779,2973,4963,6397,
%T A146556 6399,12961,14983,14985,40191,66993,114063,193233,334959,558273,
%U A146556 951999,1586673,3724815,8255985,18271887,31279473,66853647,171456753,339654927
%N A146556 Natural growth of an aliquot sequence driven by a perfect number 2^(p-1)*((2^p) - 1).
%C A146556 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 A146556 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 A146556 Mathworld, <a href="https://mathworld.wolfram.com/AliquotSequence.html">Aliquot Sequence</a>
%H A146556 Stern, <a href="http://www.lafn.org/~ax810/aliquot.htm">Aliquot Sequences from the trenches</a> [broken link?]
%F A146556 a(n) = a(n-1) + 2*(sigma(a(n-1)) - a(n-1)). - _Roderick MacPhee_, Aug 21 2012
%e A146556 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 A146556 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 A146556 NestList[2*DivisorSigma[1,#]-#&,3,40] (* _Harvey P. Dale_, Jul 16 2013 *)
%o A146556 (PARI) A146556()=a=[3];until(#a==79,a=concat(a,a[#a]+2*(sigma(a[#a])-a[#a])));a
%o A146556 (PARI) a(n)=if(n==1,3,2*sigma(a(n-1))-a(n-1)) \\ _R. K. Guy_, Jul 16 2013
%Y A146556 Cf. A000396, A008892, A215778, A216224.
%K A146556 hard,nonn
%O A146556 1,1
%A A146556 _Sergio Pimentel_, Oct 31 2008
%E A146556 More terms, as derived from p=7, driver 8128. - _R. J. Mathar_, Jan 22 2009