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A275466
Numbers which produce only positive numbers under iteration of a function X(n) = Pi(n) - C(n) + n where Pi(n) is the sum of the distinct primes dividing n and C(n) is the sum of its other divisors (including 1).
1
2, 3, 5, 9, 11, 21, 29, 35, 43, 57, 85, 123, 139, 155, 161, 203, 209, 221, 249, 259, 265, 277, 299, 323, 349, 403, 411, 517, 521, 553, 697, 755, 771, 785, 789, 949, 995, 1041, 1043, 1181, 1203, 1227, 1243, 1273, 1379, 1559, 1651, 1661, 1759, 1819, 1897, 1937, 2051, 2123, 2189, 2219
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
Sequence in context: A364059 A293036 A214125 * A317786 A019276 A097683
KEYWORD
nonn
AUTHOR
Greg Huber, Jul 28 2016
STATUS
approved