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%I #19 Jun 26 2019 10:12:47
%S 63973,126217,748657,4463641,6054985,9890881,10837321,19384289,
%T 26921089,31146661,34901461,41471521,53711113,76595761,140241361,
%U 144218341,150846961,151530401,169057801,171454321,171679561,172430401,228842209,277241401,280761481
%N Carmichael numbers n such that n-1 is not a practical number.
%C Most Carmichael numbers (odd) have practical number companions (even) that are 1 less. This sequence identifies the exceptions. However Carmichael numbers appear to have no practical number companions that are 1 more.
%H Amiram Eldar, <a href="/A265827/b265827.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1)=63973=7*13*19*37. It is a Carmichael number because its prime factors less 1 are 6, 12, 18, 36 and this set are all divisors of 63973-1=2^2*3^2*1777 that is not a practical number. It is the first such occurrence.
%t CarmichaelQ[n_] := If[Mod[n, CarmichaelLambda[n]]==1&&!PrimeQ[n], True, False]; 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[2Range[10^7]+1, CarmichaelQ[#]&&!PracticalQ[#-1] &]
%o (PARI) is_c(n) = {my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
%o is_p(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 forcomposite(n=561, 1e10, if(is_c(n) && !is_p(n-1), print1(n, ", "))) \\ _Altug Alkan_, Dec 16 2015
%Y Cf. A002997, A261614.
%K nonn
%O 1,1
%A _Frank M Jackson_, Dec 16 2015
%E More terms from _Altug Alkan_, Dec 16 2015
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