A018844 Arises from generalized Lucas-Lehmer test for primality.

4, 10, 52, 724, 970, 10084, 95050, 140452, 1956244, 9313930, 27246964, 379501252, 912670090, 5285770564, 73621286644, 89432354890, 1025412242452, 8763458109130, 14282150107684, 198924689265124

Offset: 1

Author

Robert G. Wilson v

Keywords

• easy
• nonn

Comments

Apparently this was suggested by an article by R. M. Robinson.

Starting values for Lucas-Lehmer test that result in a zero term (mod Mersenne prime Mp) after P-1 steps. - Jason Follas (jfollas_mersenne(AT)hotmail.com), Aug 01 2004

m belongs to the sequence iff m-2 is twice a square and m+2 is either three or six times a square. - René Gy, Jan 10 2019

A006242 is a subsequence. - Davide Rotondo, Oct 21 2024

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1370
• D. H. Lehmer, An Extended Theory of Lucas' Functions, Ann. Math. 31 (1930), 419-448. See p. 445.
• E. L. Roettger and H. C. Williams, Some Remarks Concerning the Lucas-Lehmer Primality Test, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.5. See pp. 3, 13, 24.
• Herb Savage et al., Re: Mersenne: starting values for LL-test.

Crossrefs

Cf. A006242, A206257.

Programs

PARI

listUpTo(n)=a=List([4,52]);while(1,m=14*a[#a]-a[#a-1];m>n&&break();listput(a,m));b=List([10,970]);while(1,m=98*b[#b]-b[#b-1];m>n&&break();listput(b,m));setunion(Set(a),Set(b)) \\ Jeppe Stig Nielsen, Aug 03 2020

Formula

Union of sequences a_1=4, a_2=52, a_{n}=14*a_{n-1} - a_{n-2} and b_1=10, b_2=970, b_{n}=98*b_{n-1} - b_{n-2}.

a[1]=14 (mod Mp), a[2]=52 (mod Mp), a[n]=(14*a[n-1]-a[n-2]) (mod Mp). - Jason Follas (jfollas_mersenne(AT)hotmail.com), Aug 01 2004

Though originally noted as the union of two sequences, when the first sequence (14*a[n-1]-a[n-2]) is evaluated modulo a Mersenne prime, the terms of the second sequence (98*b[n-1]-b[n-2]) will occur naturally (just not in numerical order). - Jason Follas (jfollas_mersenne(AT)hotmail.com), Aug 01 2004

a(n) = sqrt(A206257(n) + 2). - Arkadiusz Wesolowski, Feb 08 2012