Maurice Mischler - cp's OEIS frontend

User: Maurice Mischler

Maurice Mischler has authored 2 sequences.

A086208 Prime numbers p with the property that the sequence p-12, p-23, p-34, p-45, ..., consists only of primes and a final 1.

3, 7, 13, 43, 73

Offset: 0

Maurice Mischler (maurice.mischler(AT)ima.unil.ch), Aug 27 2003

Related to the fact that the list 5, 8, 13, 29, 53, 173, 293 gives the only positive discriminants of class number 1 of the form j^2+1.

If we allow the first number that we subtract to be greater than 2, the next sequence after 73, 71, 67, 61,... is 2003, 1831, 1657, 1481, 1303, 1123, 941, 757, 571, 383, 193, 1: start with 2003 and subtract 172,174,176, etc., which is an amusing property of 2003.

			For example 73 71 67 61 53 43 31 17 1. If the GRH is true this is the last example.

Definition corrected by Harvey P. Dale, Jul 25 2015

A086695 a(n) = 100^n - 10^n - 1.

Original entry on oeis.org

89, 9899, 998999, 99989999, 9999899999, 999998999999, 99999989999999, 9999999899999999, 999999998999999999, 99999999989999999999, 9999999999899999999999, 999999999998999999999999, 99999999999989999999999999, 9999999999999899999999999999

Offset: 1

Maurice Mischler (maurice.mischler(AT)ima.unil.ch), Sep 12 2003

Digits of the inverses of these numbers give the Fibonacci numbers. More precisely, the digits of 1/(10^(2*n)-10^n-1) give the Fibonacci numbers up to 10^n.
More generally, if x_1, x_2, x_n=x_(n-1)-x_(n-2) is any Lucas sequence, then the digits of the numbers (x_1*10^n-(x_1-x_2))/(10^(2*n)-10^n-1) give the x_n up to 10^n.
1/a(n) = Sum_{i>=1} A000045(i-1)/10^(n*i) (see Long paper). - Michel Marcus, May 01 2013

C. T. Long, The Decimal Expansion of 1/89 and Related Results, The Fibonacci Quarterly, Volume 19, Number 1, February 1981
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A000045.

Table[100^n-10^n-1,{n,20}] (* or *) LinearRecurrence[{111,-1110,1000},{89,9899,998999},20] (* Harvey P. Dale, Nov 16 2023 *)

a(n)=100^n-10^n-1 \\ Charles R Greathouse IV, May 01 2013

a(n) = 10^(2*n) - 10^n - 1.

Offset corrected by Jon E. Schoenfield, Jun 17 2018

cp's OEIS Frontend

User: Maurice Mischler

A086208 Prime numbers p with the property that the sequence p-12, p-23, p-34, p-45, ..., consists only of primes and a final 1.

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A086695 a(n) = 100^n - 10^n - 1.

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cp's OEIS Frontend

User: Maurice Mischler

A086208 Prime numbers p with the property that the sequence p-1*2, p-2*3, p-3*4, p-4*5, ..., consists only of primes and a final 1.

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Author

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Examples

Extensions

A086695 a(n) = 100^n - 10^n - 1.

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A086208 Prime numbers p with the property that the sequence p-12, p-23, p-34, p-45, ..., consists only of primes and a final 1.