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

Author

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

Keywords

• nonn
• base

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

Examples

			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.
		

Crossrefs

Cf. A003010, A177874, A000043.

Programs

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];

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]]

Extensions

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