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 A161918 Numbers n such that the sum of the divisors minus the sum of the prime factors (counted with multiplicity) is equal to n+1. 3

%I

%S 6,8,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,62,65,69,74,77,

%T 82,85,86,87,91,93,94,95,106,111,115,118,119,122,123,129,133,134,141,

%U 142,143,145,146,155,158,159,161,166,177,178,183,185,187,194,201,202,203

%N Numbers n such that the sum of the divisors minus the sum of the prime factors (counted with multiplicity) is equal to n+1.

%C Equals A006881 union {8}. - _Franklin T. Adams-Watters_, Jun 26 2009

%H Jayanta Basu, <a href="/A161918/b161918.txt">Table of n, a(n) for n = 1..10000</a>

%e n=21: Sum_divisors (1,3,7,21) = 32; Sum_prime_factors (3,7) = 10 -> 32-10 = 22. n=55: Sum_divisors (1,5,11,55) = 72; Sum_prime_factors (5,11) = 16 -> 72-16 = 56.

%p with(numtheory); P:=proc(i) local b,c,j,s,n; for n from 2 by 1 to i do b:=(convert(ifactors(n),`+`)-1); c:=nops(b); j:=0; s:=0; for j from c by -1 to 1 do s:=s+convert(b[j],`*`); od; if n=sigma(n)-s-1 then print(n); fi; od; end: P(500);

%t Select[Range[203], DivisorSigma[1, #] - Total[Times @@@ FactorInteger[#]] == # + 1 &] (* _Jayanta Basu_, Aug 11 2013 *)

%o (PARI from _M. F. Hasler_):

%o isA161918(n)={ n+1 == sigma(n)-(n=factor(n))[,1]~*n[,2] }

%o for(n=1,500, isA161918(n)&print1(n","))

%Y Cf. A161917, A006881, A151797, A030229.

%K easy,nonn

%O 1,1

%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Jun 23 2009

%E Edited by _N. J. A. Sloane_, Jun 27 2009 incorporating suggestions from _R. J. Mathar_, _M. F. Hasler_, _Benoit Jubin_ and others.

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