OFFSET
1,1
COMMENTS
The first four survivors, a(1) = 2, a(2) = 3, a(3) = 5, and a(4) = 9, are the members of the loop 2,3,5,9,2,3,5,..., apparently the only loop possible under the iteration (empirical observation).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Kyle Kawagoe and Greg Huber, An iteration based on prime and composite factors, arXiv:1608.06593 [math.NT], 2016.
EXAMPLE
a(1) = 2 since X(2) = 2-1+2 = 3, and the iteration has entered the 2,3,5,9 loop.
a(5) = 11 is the fifth survivor, since X(11) = 11-1+11 = 21, X(21) = (3+7)-(21+1)+21 = 9, and the iteration has entered the 9,2,3,5 loop.
1 is not a member of the sequence, since it does not map to a positive integer.
MATHEMATICA
A275466Q[n_] := Catch[NestWhile[2 Total[FactorInteger[#][[All, 1]]] - DivisorSigma[1, #] + # &, n, Which[# == 9, Throw[True], # <= 1, Throw[False], True, True] &]]; Select[Range[500], A275466Q] (* JungHwan Min, Jul 29 2016 *)
PROG
(PARI) X(n, f=factor(n))=n + 2*sum(i=1, #f~, f[i, 1]) - sigma(f)
is(n)=my(t=X(n), h); if(t<1, return(0)); h=X(t); if(h<1, return(0)); while(t!=h, t=X(t); h=X(h); if(h<1, return(0)); h=X(h); if(h<1, return(0))); 1 \\ Charles R Greathouse IV, Jul 28 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Greg Huber, Jul 28 2016
STATUS
approved