A135928 -id:A135928 - cp's OEIS frontend

A135927 a(n) = a(n-1)^2 - 2 with a(1) = 10.

10, 98, 9602, 92198402, 8500545331353602, 72259270930397519221389558374402, 5221402235392591963136699520829303150191924374488750728808857602

Offset: 1

Ant King, Dec 07 2007

This is the Lucas-Lehmer sequence with starting value u(1) = 10 and the position of the zeros when it is reduced mod(2^p - 1) also gives the position of the Mersenne primes. As we have started with n = 1, these will occupy the (p - 1)th positions in the sequence. For example, the first 12 terms mod(2^13 - 1) are 10, 98, 1411, 506, 2113, 672, 1077, 4996, 2037, 4721, 128, 0 and hence 8191 is a Mersenne prime. The radicals in the above closed forms are the solutions to x^2 - 10x + 1 = 0.

			a(4) = 2*cosh(2^3*log(5 + 2*sqrt(6))) = 92198402.

Gabriel Klambauer, Summation of Series, Amer. Math. Monthly, Vol. 87, No. 2 (Feb., 1980), pp. 128-130.
Raphael M. Robinson, Mersenne and Fermat Numbers, Proceedings of the American Mathematical Society, Vol. 5, No. 5. (October 1954), pp. 842-846.
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 p. 3.
Wikipedia, Engel expansion

Cf. A000668, A000043, A003010, A095847, A001566, A135928, A246723.

a[1] = 10; a[n_] := a[n] = a[n - 1]^2 - 2; a[#]&/@Range[7]

A135927 = [10]
for n in range(1, 8): A135927.append(A135927[-1]**2-2)
print(A135927) # Karl-Heinz Hofmann, Feb 01 2022

a(n) = 2*cosh(2^(n-1)*log(5 + 2*sqrt(6))) = exp(2^(n-1)*log(5 + 2*sqrt(6))) + exp(2^(n-1)*log(5 - 2*sqrt(6))) = (5 + 2*sqrt(6))^(2^(n-1)) + (5 - 2*sqrt(6))^(2^(n-1)) = ceiling(exp(2^(n-1)*log(5 + 2*sqrt(6)))) = ceiling((5 + 2*sqrt(6))^(2^(n-1))).
From Peter Bala, Feb 01 2022: (Start)
Product_{n >= 1} (1 + 2/a(n)) = (1/2)*sqrt(6); Product_{n >= 1} (1 - 1/a(n)) = (4/11)*sqrt(6).
Engel expansion of 5 - sqrt(24) = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + .... See Klambauer, p. 130. (End)

A136188 Digital roots of the Fermat numbers in A000215(n).

Original entry on oeis.org

3, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8

Offset: 0

Ant King, Dec 24 2007

As 2^(2^n)+1=5 (mod 9) for odd values of n and 2^(2^n)+1=8 (mod 9) for even values of n>0, it follows that the digital roots of the Fermat numbers form a cyclic sequence, with the 5's corresponding to odd values of n and the 8's to even values of n.

Decimal expansion of 71/198. - Enrique Pérez Herrero, Nov 13 2021

			2^(2^3) + 1 = 257. This has digital root 5 and hence a(3) = 5.

I. Izmirli, On Some Properties of Digital Roots, Advances in Pure Mathematics, Vol. 4 No. 6 (2014), Article ID:47285.
Eric Weisstein's World of Mathematics, Digital Root.
Eric Weisstein's World of Mathematics, Fermat Number.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000215, A010888, A135928.

Essentially the same as A010719.

FermatNumber[n_]:=2^(2^n)+1;DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[ # ]&,n];DigitalRoot/@(FermatNumber[ # ] &/@Range[0,25])

a(n)=if(n,if(n%2,5,8),3) \\ Charles R Greathouse IV, May 01 2016

a(n) = A010888(A000215(n)).

cp's OEIS Frontend

A135927 a(n) = a(n-1)^2 - 2 with a(1) = 10.

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