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).

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

Adjacent sequences: A275463 A275464 A275465 * A275467 A275468 A275469

Keyword

nonn

Author

Greg Huber, Jul 28 2016

Status

approved