A177875 The number of decimal digits in Lucas-Lehmer numbers A003010(k-2) divisible by 2^k-1.

2, 5, 19, 1172, 18742, 74967, 307062002, 329705313529178423, 88504596182827979077122168, 23200948861751257747193113585514, 24327958153659686843520766271043070385

Offset

1,1

Comments

The values of k are 3, 5, 7, 13, ..., the odd Mersenne prime exponents A000043.

A003010(n) has A177874(n) decimal digits.

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

Links

Table of n, a(n) for n=1..11.

Example

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.
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.

Mathematica

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]]

Prog

(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];

Crossrefs

Cf. A003010, A177874, A000043.

Sequence in context: A270398 A269997 A270547 * A187602 A260140 A270556

Adjacent sequences: A177872 A177873 A177874 * A177876 A177877 A177878

Keyword

nonn,base

Author

G. L. Honaker, Jr., Dec 13 2010

Extensions

a(7)-a(11) from D. S. McNeil, Dec 13 2010

Status

approved