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A221743
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Numbers k such that (6*k+1)*(12*k+1)*(18*k+1) is a Carmichael number which is the product of four prime numbers.
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2
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5, 11, 15, 61, 85, 115, 455, 661, 700, 805, 920, 1225, 1326, 1910, 2961, 4935, 5425, 6565, 8175, 10885, 11375, 12155, 13230, 18315, 37800, 39325, 45325, 59726, 69440, 99645, 113120, 121365, 129850, 144685, 211945, 353465, 378940, 389896, 392625
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OFFSET
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1,1
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LINKS
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MAPLE
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with(numtheory); P:=proc(q)local a, b, k, ok, n;
for n from 0 to q do a:=(6*n+1)*(12*n+1)*(18*n+1); b:=ifactors(a)[2];
if issqrfree(a) and nops(b)=4 then ok:=1;
for k from 1 to 4 do if not type((a-1)/(b[k][1]-1), integer) then ok:=0;
break; fi; od; if ok=1 then print(n); fi;
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MATHEMATICA
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IsCarmichaelQ[n_] := Module[{f}, If[EvenQ[n] || PrimeQ[n], False, f = Transpose[FactorInteger[n]][[1]]; Union[Mod[n-1, f-1]] == {0}]]; n = 0; t = {}; While[Length[t] < 39, n++; c = (6*n + 1)*(12*n + 1)*(18*n + 1); If[SquareFreeQ[c] && Length[FactorInteger[c]] == 4 && IsCarmichaelQ[c], AppendTo[t, n]]]; t (* T. D. Noe, Jan 23 2013 *)
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PROG
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(Magma) [n: n in [1..4*10^5] | #PrimeDivisors(c) eq 4 and IsOne(c mod CarmichaelLambda(c)) where c is (6*n+1)*(12*n+1)*(18*n+1)];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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