A018844 -id:A018844 - cp's OEIS frontend

A206257 Values of S(1) such that any Mersenne prime with an odd exponent p divides S(p-2), where S(n) == S(n-1)^2 - 2 (mod M(p)).

14, 98, 2702, 524174, 940898, 101687054, 9034502498, 19726764302, 3826890587534, 86749292044898, 742397047217294, 144021200269567502, 832966693180608098, 27939370455248878094, 5420093847118012782734, 7998146101170906912098, 1051470266970439230972302

Offset: 1

Arkadiusz Wesolowski, Feb 05 2012

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..150
Wikipedia, Lucas-Lehmer primality test

nn = 17; t1 = LinearRecurrence[{194, -1}, {14, 2702}, nn]; t2 = LinearRecurrence[{9602, -1}, {98, 940898}, nn]; t3 = Select[t2, # < t1[[-1]]&]; Union[t1, t3]

Union of sequences a(0) = 14, a(1) = 2702; a(n) = 194*a(n-1) - a(n-2) and b(0) = 98, b(1) = 940898; b(n) = 9602*b(n-1) - b(n-2).

a(n) = A018844(n)^2 - 2.

A304177 Union of sequences b and c defined by: b(1)=8, b(2)=488, b(n)=62b(n-1) - b(n-2) and c(1)=22, c(2)=10582, c(n)=482c(n-1) - c(n-2).

Original entry on oeis.org

8, 22, 488, 10582, 30248, 1874888, 5100502, 116212808, 2458431382, 7203319208, 446489578088, 1184958825622, 27675150522248, 571147695518422, 1715412842801288, 106327921103157608, 275292004281053782, 6590615695552970408, 132690174915772404502, 408511845203181007688

Offset: 1

Pedja Terzic, May 07 2018

Conjecture: Each member of this sequence can be used as an initial value for Inkeri's primality test for Fermat numbers.

Inkeri's primality test for Fermat numbers: Fermat's number F_{m}=2^2^m+1 (m => 2) is prime if and only if F_{m} divides the term v_{2^m-2} of the series v_{0}=8 , v_{i}=(v_{i-1})^2-2 .

K. Inkeri, Tests for primality, Ann. Acad. Sci. Fenn., A I 279, 119 (1960).

Pedja Terzic, Initial values of Inkeri's primality test for Fermat numbers, Math StackExchange, May 2018.

Cf. A018844, A000215.

b=RecurrenceTable[{a[1]==8,a[2]==488,a[n]==62a[n-1]-a[n-2]},a,{n,12}]; c= RecurrenceTable[{a[1]==22,a[2]==10582,a[n]==482a[n-1]-a[n-2]},a,{n,12}]; Join[ b,c]//Union (* Harvey P. Dale, May 05 2022 *)

InitialValues(n)= {l=[8,22,488,10582];b1=8;b2=488;i=3;while(i<=n,b=62*b2-b1;l=concat(l,b);b1=b2;b2=b;i++);c1=22;c2=10582;j=3;while(j<=n,c=482*c2-c1;l=concat(l,c);c1=c2;c2=c;j++);print(vecsort(l))}

cp's OEIS Frontend

A206257 Values of S(1) such that any Mersenne prime with an odd exponent p divides S(p-2), where S(n) == S(n-1)^2 - 2 (mod M(p)).

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A304177 Union of sequences b and c defined by: b(1)=8, b(2)=488, b(n)=62b(n-1) - b(n-2) and c(1)=22, c(2)=10582, c(n)=482c(n-1) - c(n-2).

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cp's OEIS Frontend

A206257 Values of S(1) such that any Mersenne prime with an odd exponent p divides S(p-2), where S(n) == S(n-1)^2 - 2 (mod M(p)).

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A304177 Union of sequences b and c defined by: b(1)=8, b(2)=488, b(n)=62*b(n-1) - b(n-2) and c(1)=22, c(2)=10582, c(n)=482*c(n-1) - c(n-2).

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Programs

Mathematica

PARI

A304177 Union of sequences b and c defined by: b(1)=8, b(2)=488, b(n)=62b(n-1) - b(n-2) and c(1)=22, c(2)=10582, c(n)=482c(n-1) - c(n-2).