A010709 -id:A010709 - cp's OEIS frontend

A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.

46, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

More generally, the ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2, k = 1, 2, 3,... is floor(phi^(2*k + 1))/(1 - x), and for the continued fraction expansion of phi^(2*k) is (floor(phi^(2*k)) + x - x^2)/(1 - x^2).

			phi^8 = (47 + 21*sqrt(5))/2 = 46 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/...)))))).

Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden ratio
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A001622.

Cf. continued fraction expansion of phi^k: A000012 (k = 1), A054977 (k = 2), A010709 (k = 3), A176260 (k = 4, for n>0), A010850 (k = 5), A040071 (k = 6, for n>0), A010868 (k = 7), this sequence (k = 8).

[46] cat &cat [[1, 45]^^50]; // Vincenzo Librandi, Jan 13 2016

ContinuedFraction[(47 + 21 Sqrt[5])/2, 82]

G.f.: (46 + x - x^2)/(1 - x^2).

a(n) = 23 + 22*(-1)^n for n>0. - Bruno Berselli, Jan 18 2016

A267649 a(0) = a(1) = 2 then a(n) = 4 for n>=2.

Original entry on oeis.org

2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Natan Arie Consigli, Jan 19 2016

Decimal expansion of 101/450.
Also list of smallest n-composites.
A hyperoperator aggregation b[n]c is n-composite if b,c are positive non-right-identity elements.
The identity elements are:
Hyper-0 (zeration): none.
Hyper-1 (addition): 0.
Hyper-2 (multiplication): 1.
Hyper-3 (exponentiation): 1.
Hyper-n (n>2): 1.
For more information on hyperoperations see A054871.
Essentially the same as A255176, A151798, A123932, A113311, A040002 and A010709. - R. J. Mathar, May 25 2023
Continued fraction expansion of 2 + sqrt(1/5) = 2 + sqrt(5)/5. - Elmo R. Oliveira, Aug 06 2024

			a(0) = 2 because 1 is the smallest non-identity element in zeration and 1[0]1=2;
a(1) = 2 because 1 is the smallest non-identity element in addition and 1[1]1=2;
a(2) = 4 because 2 is the smallest non-identity element in multiplication and 2[2]2=4;
a(3) = 4 because 2 is the smallest non-identity element in exponentiation and 2[2]2=4;
a(4) = 4 because 2 is the smallest non-identity element in titration and 2[2]2=4;
Etc.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000027 (1-composites), A002808 (composites), A267647 (3-composites), A097374 (4-composites).

a(n) = a[n]b where a,b are the positive smallest non-right-identity elements.
From Elmo R. Oliveira, Aug 06 2024: (Start)
G.f.: 4/(1 - x) - 2*(1 + x).
E.g.f.: 4*exp(x) - 2*(1 + x). (End)

A361683 a(n) is the least k such that tau(k) divides sigma_n(k) but not sigma(k), or -1 if no such k exists.

Original entry on oeis.org

4, 64, 4, 7168, 4, 606528, 4, 64, 4, 4194304, 4

Offset: 2

Mohammed Yaseen, Mar 20 2023

a(13) <= 31525197391593472. - David A. Corneth, Mar 20 2023
From Thomas Scheuerle, Mar 22 2023: (Start)
a(17) <= 15211807202738752817960438464512 and a(19) <= 2^190*11.
Conjecture: a(n) is of the form 2^b*p1^c*p2^d*...*pk^j with b > 0 and A020639(n) divides b*(c+1)*(d+1)*...*(j+1). (p1, p2, ..., pk are distinct odd prime numbers). (End)

Cf. A003601, A020486, A046839, A046840.
Cf. A000005, A000203, A001157, A001158.
Cf. A020639, A010709 (bisection).

a[n_] := Module[{k = 1, d}, While[Divisible[DivisorSigma[1, k], (d = DivisorSigma[0, k])] || !Divisible[DivisorSigma[n, k], d], k++]; k]; Array[a, 11, 2] (* Amiram Eldar, Mar 20 2023 *)

isok(k, n) = my(f=factor(k), nd=numdiv(f)); (sigma(f) % nd) && !(sigma(f,n) % nd);
a(n) = my(k=1); while (!isok(k,n), k++); k; \\ Michel Marcus, Mar 20 2023

a(2*m) = 4 for m >= 1.
a(6*m-3) = 64 for m >= 1.
From Thomas Scheuerle, Mar 22 2023: (Start)
a(m) <= a(A020639(m)) if a(A020639(m)) <> -1.
Conjecture: For primes q > p, a(q) > a(p). If true, we could replace "<=" with "=" in the above formula. (End)

A376324 (1/6) times obverse convolution (4)**(2^n + 1); see Comments.

Original entry on oeis.org

1, 7, 63, 819, 17199, 636363, 43909047, 5839903251, 1524214748511, 788019024980187, 810871576704612423, 1664719346974569304419, 6827014041942708717422319, 55961034101804383356710748843, 917145387894472038833132462787927

Offset: 0

Clark Kimberling, Sep 20 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

Cf. A000051, A010709, A168614, A374848, A375053, A375054.

s[n_] := 4; t[n_] := 2^n + 1;
u[n_] := (1/6) Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
(* or *)
Table[2^(n*(n+1)/2 - 1) * QPochhammer[-5, 1/2, n+1]/3, {n, 0, 15}] (* Vaclav Kotesovec, Sep 20 2024 *)

a(n) = a(n-1)*A168614(n) for n>=1.

cp's OEIS Frontend

A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.

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A267649 a(0) = a(1) = 2 then a(n) = 4 for n>=2.

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A361683 a(n) is the least k such that tau(k) divides sigma_n(k) but not sigma(k), or -1 if no such k exists.

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A376324 (1/6) times obverse convolution (4)**(2^n + 1); see Comments.

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