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A174164 Numbers n such that 1 = abs(sum{p-1|p is prime and divisor of n} - product{p-1|p is prime and divisor of n}).

%I A174164 #15 Sep 17 2020 06:40:35
%S A174164 6,10,12,14,18,20,22,24,26,28,30,34,36,38,40,44,46,48,50,52,54,56,58,
%T A174164 60,62,68,72,74,76,80,82,86,88,90,92,94,96,98,100,104,106,108,112,116,
%U A174164 118,120,122,124,134,136,142,144,146,148,150,152,158,160,162,164,166,172
%N A174164 Numbers n such that 1 = abs(sum{p-1|p is prime and divisor of n} - product{p-1|p is prime and divisor of n}).
%H A174164 Robert Israel, <a href="/A174164/b174164.txt">Table of n, a(n) for n = 1..10000</a>
%e A174164 6 is a term because 6=2*3 and 1=abs((2-1)+(3-1)-(2-1)*(3-1)).
%e A174164 10 is a term because 10=2*5 and 1=abs((2-1)+(5-1)-(2-1)*(5-1)).
%p A174164 From _R. J. Mathar_, Apr 26 2010: (Start)
%p A174164 A055631 := proc(n) add(d-1, d= numtheory[factorset](n) ) ; end proc:
%p A174164 A173557 := proc(n) mul(d-1, d= numtheory[factorset](n) ) ; end proc:
%p A174164 isA174164 := proc(n) A055631(n)-A173557(n) ; abs(%) = 1 ; end proc:
%p A174164 for n from 2 to 200 do if isA174164(n) then printf("%d,",n) ; end if; end do: (End)
%t A174164 filterQ[n_] := With[{pp = FactorInteger[n][[All, 1]]}, 1 == Abs[Total[pp-1] - Times @@ (pp-1)]];
%t A174164 Select[Range[200], filterQ] (* _Jean-François Alcover_, Sep 17 2020 *)
%Y A174164 Cf. A055631, A173557.
%Y A174164 Union of A100367 and A143207.
%K A174164 nonn
%O A174164 1,1
%A A174164 _Juri-Stepan Gerasimov_, Mar 10 2010
%E A174164 Corrected (53 replaced by 52, 90 and 120 inserted) by _R. J. Mathar_, Apr 26 2010