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%I #23 Sep 08 2022 08:45:53
%S 2,5,19,1172,18742,74967,307062002,329705313529178423,
%T 88504596182827979077122168,23200948861751257747193113585514,
%U 24327958153659686843520766271043070385
%N The number of decimal digits in Lucas-Lehmer numbers A003010(k-2) divisible by 2^k-1.
%C The values of k are 3, 5, 7, 13, ..., the odd Mersenne prime exponents A000043.
%C A003010(n) has A177874(n) decimal digits.
%C The larger terms can be computed by combining techniques from both integer and real arithmetic. The values of k for which A003010(k-2) is divisible by 2^k-1 are found from computing A003010 recursively mod 2^k-1. Unfortunately this gives no information on the number of decimal digits of A003010(k-2), i.e. A177874(k-2). To determine this, we use arbitrary-precision interval arithmetic -- in which we quickly lose information about divisibility -- to place bounds on the size of A003010(k-2) and find d such that 10^(d-1) <= A003010(k-2) < 10^d. - _D. S. McNeil_, Dec 14 2010
%e k = 3: Lucas-Lehmer number A003010(1) = 14 is divisible by 2^3-1 = 7 and has 2 decimal digits. Hence A177874(1) = 2 is in the sequence.
%e k = 7: Lucas-Lehmer number A003010(5) = 2005956546822746114 is divisible by 2^7-1 = 127 and has 19 decimal digits. Hence A177874(5) = 19 is in the sequence.
%t a=Sqrt[6]; Reap[Do[a=a^2-2; If[Mod[a, 2^(n+1)-1]==0, Sow[Length[IntegerDigits[a]]]], {n, 26}]][[2, 1]]
%o (Magma) T:=[ n eq 1 select 4 else Self(n-1)^2-2: n in [1..24] ]; a003010:=func< n | T[n+1] >; a177874:=func< n | #Intseq(a003010(n)) >; [ a177874(n): n in [0..#T-1] | a003010(n) mod (2^(n+2)-1) eq 0];
%Y Cf. A003010, A177874, A000043.
%K nonn,base
%O 1,1
%A _G. L. Honaker, Jr._, Dec 13 2010
%E a(7)-a(11) from _D. S. McNeil_, Dec 13 2010
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