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A182087 Carmichael numbers of the form C = (30n-p)*(60n-(2p+1))*(90n-(3p+2)), where n is a natural number and p, 2p+1, 3p+2 are all three prime numbers.

%I A182087 #26 Feb 16 2025 08:33:13
%S A182087 1729,172081,294409,1773289,4463641,56052361,118901521,172947529,
%T A182087 216821881,228842209,295643089,798770161,1150270849,1299963601,
%U A182087 1504651681,1976295241,2301745249,9624742921,11346205609,13079177569
%N A182087 Carmichael numbers of the form C = (30n-p)*(60n-(2p+1))*(90n-(3p+2)), where n is a natural number and p, 2p+1, 3p+2 are all three prime numbers.
%C A182087 These numbers can be reduced to only two possible forms: C =(30n-23)*(60n-47)*(90n-71) or C = (30n-29)*(60n-59)*(90n-89). In the first form, for the particular case when 30n-23,60n-47 and 90n-71 are all three prime numbers, we obtain the Chernick numbers of the form 10m+1 (for k = 5n-4 we have C = (6k+1)*(12k+1)*(18k+1)). In the second form,  for the particular case when 30n-29,60n-59 and 90n-89 are all three prime numbers, we obtain the Chernick numbers of the form 10m+9 (for k = 5n-5 we have C = (6k+1)*(12k+1)*(18k+1)).
%C A182087 So the Chernick numbers can be divided into two categories: Chernick numbers of the form (30n+7)*(60n+13)*(90n+19) and Chernick numbers of the form (30n+1)*(60n+1)*(90n+1).
%H A182087 Charles R Greathouse IV, <a href="/A182087/b182087.txt">Table of n, a(n) for n = 1..10000</a>
%H A182087 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>.
%o A182087 (PARI) list(lim)={
%o A182087     my(v=List(),f);
%o A182087     for(k=1,round(solve(x=(lim/162000)^(1/3),lim^(1/3),(30*x-23)*(60*x-47)*(90*x-71)-lim)),
%o A182087         n=(30*k-23)*(60*k-47)*(90*k-71)-1;
%o A182087         f=factor(30*k-23);
%o A182087         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182087         f=factor(60*k-47);
%o A182087         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182087         f=factor(90*k-71);
%o A182087         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182087         listput(v,n+1)
%o A182087     );
%o A182087     for(k=2,round(solve(x=(lim/162000)^(1/3),lim^(1/3),(30*x-29)*(60*x-59)*(90*x-89)-lim)),
%o A182087         n=(30*k-29)*(60*k-59)*(90*k-89)-1;
%o A182087         f=factor(30*k-29);
%o A182087         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182087         f=factor(60*k-59);
%o A182087         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182087         f=factor(90*k-89);
%o A182087         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182087         listput(v,n+1)
%o A182087     );
%o A182087     vecsort(Vec(v))
%o A182087 }; \\ _Charles R Greathouse IV_, Oct 02 2012
%Y A182087 Cf. A033502, A206347.
%K A182087 nonn
%O A182087 1,1
%A A182087 _Marius Coman_, Apr 11 2012