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%I #15 Dec 01 2015 15:42:22
%S 9,10,14,15,21,22,25,26,27,33,34,35,38,39,44,45,46,49,50,51,52,55,57,
%T 58,62,63,65,68,69,70,74,75,76,77,81,82,85,86,87,91,92,93,94,95,98,99,
%U 102,105,106,110,111
%N Numbers that are neither prime (A000040) nor practical (A005153).
%e a(5)=21 and it is neither prime nor practical. It is the 5th such occurrence.
%t PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Select[Range[1, 1000], ! PracticalQ[#] && ! PrimeQ[#] &] (* using _T. D. Noe_'s program A005153 *)
%o (PARI) isok(n) = bittest(n, 0) && return(n==1); my(P=1); n && !for(i=2, #n=factor(n)~, n[1, i]>1+(P*=sigma(n[1, i-1]^n[2, i-1])) && return) ;
%o for(n=1, 200, if(!isok(n) && !isprime(n), print1(n, ", "))) \\ _Altug Alkan_, Nov 19 2015
%Y Cf. A000040, A005153, A225223.
%K nonn
%O 1,1
%A _Frank M Jackson_, Nov 18 2015
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