A055500 -id:A055500 - cp's OEIS frontend

A065435 a(3) = 2, a(4) = 3; for n > 4, a(n) = {a(n-2)}+{a(n-1)}, where {a} means largest prime <= a.

2, 3, 5, 8, 12, 18, 28, 40, 60, 96, 148, 228, 366, 586, 936, 1506, 2428, 3922, 6342, 10256, 16590, 26826, 43394, 70212, 113598, 183798, 297388, 481174, 778548, 1259712, 2038242, 3297918, 5336130, 8634042, 13970112, 22604076, 36574162

Offset: 3

Bodo Zinser, Nov 17 2001

			a(9) = 28 because 11+17 = 28 and 11 largest prime <= a(7) = 12 and 17 is largest prime <= a(8) = 18

Harry J. Smith, Table of n, a(n) for n = 3..300

Cf. A055500, A000040, A000045, A007917, A055500.

a065435 n = a065435_list !! (n-3)
a065435_list = 2 : 3 : zipWith (+) xs (tail xs) where
               xs = map (a007917 . fromInteger) a065435_list
-- Reinhard Zumkeller, Aug 10 2012

PrevPrim[n_] := Block[ {k = n}, While[ !PrimeQ[k], k-- ]; Return[k]]; a[3] = 2; a[4] = 3; a[n_] := a[n] = PrevPrim[ a[n - 1]] + PrevPrim[ a[n - 2]]; Table[ a[n], {n, 3, 45} ]
np[n_]:=If[PrimeQ[n],n,NextPrime[n,-1]]; Transpose[NestList[{Last[#], np[Last[#]]+np[First[#]]}&,{2,3},40]][[1]] (* Harvey P. Dale, Oct 01 2011 *)

for (n=3, 300, if (n>4, a=precprime(a2) + precprime(a1); a2=a1; a1=a, if (n==4, a=a1=3, a=a2=2)); write("b065435.txt", n, " ", a) ) \\ Harry J. Smith, Oct 18 2009

a(n) = A007917(a(n-2)) + A007917(a(n-1)). - Jonathan Vos Post, Jul 10 2008

More terms from Robert G. Wilson v, Nov 19 2001

Definition corrected by Harry J. Smith, Oct 18 2009

A071353 First term of the continued fraction expansion of (3/2)^n.

Original entry on oeis.org

2, 4, 2, 16, 1, 2, 11, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 8, 5, 1, 7, 1, 25, 16, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 1, 1, 2, 7, 4, 3, 2, 4, 1, 3, 1, 3, 1, 1, 1, 2, 10, 1, 2, 4, 1, 4, 2, 1, 3, 2, 14, 9, 6, 1, 11, 1, 1, 2, 1, 1, 2, 6, 1, 12, 1, 1, 2, 1, 2, 19, 12, 8, 1, 89, 59, 1, 3

Offset: 1

Paul D. Hanna, Jun 10 2002

If uniformly distributed, then the average of the reciprocal terms of this sequence is 1/2.

"Pisot and Vijayaraghavan proved that (3/2)^n has infinitely many accumulation points, i.e. infinitely many convergent subsequences with distinct limits. The sequence is believed to be uniformly distributed, but no one has even proved that it is dense in [0,1]." - S. R. Finch.

			a(7) = 11 since floor(1/frac(3^7/2^7)) = floor(1/.0859375) = 11.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 192-199.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Steven R. Finch, Powers of 3/2 Modulo One [From Steven Finch, Apr 20 2019]
Steven R. Finch, Non-Ideal Waring's Problem [From Steven Finch, Apr 20 2019]
Jeff Lagarias, 3x+1 Problem
C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scuola Norm. Sup. Pisa, 7 (1938), 205-248.
T. Vijayaraghavan, On the fractional parts of the powers of a number (I), J. London Math. Soc. 15 (1940) 159-160.

Cf. A055500, A071291.

Table[Floor[1/FractionalPart[(3/2)^n]], {n, 1, 100}] (* G. C. Greubel, Apr 18 2017 *)

a(n) = contfrac((3/2)^n)[2] \\ Michel Marcus, Aug 01 2013

a(n) = floor(1/frac((3/2)^n)).

A345020 a(0) = a(1) = 1, a(n) = largest natural number m <= a(n-1) + a(n-2) where gcd(m,a(k)) = 1 for all 1 < k <= n-1.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 17, 23, 37, 59, 89, 139, 227, 361, 587, 947, 1531, 2477, 4007, 6481, 10487, 16963, 27449, 44393, 71837, 116227, 188063, 304289, 492343, 796627, 1288967, 2085593, 3374557, 5460139, 8834689, 14294827, 23129507, 37424333, 60553837, 97978169

Offset: 0

Amrit Awasthi, Jun 05 2021

nonn

First differs from A055500 at a(14).

			a(5) = 7 because 7 is the largest number less than or equal to a(4) + a(3) = 8 which is coprime to all the previous terms of sequence.

Robert Israel, Table of n, a(n) for n = 0..4781

Cf. A055500.

A[0]:= 1:
A[1]:= 1: P:= 1:
for n from 2 to 100 do
  for k from A[n-2]+A[n-1] by -1 do
    if igcd(k,P) = 1 then break fi
  od;
  A[n]:= k;
  P:= P*k;
od:
convert(A,list); # Robert Israel, Oct 23 2024

a[0] = a[1] = 1; a[n_] := a[n] = Module[{k = a[n - 1] + a[n - 2]}, While[! AllTrue[Range[2, n - 2], CoprimeQ[a[#], k] &], k--]; k]; Array[a, 50, 0] (* Amiram Eldar, Jun 05 2021 *)

A126273 a(0) = a(1) = a(2) = 1, a(n) = largest prime <= a(n-1) + a(n-2) + a(n-3).

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 13, 23, 43, 79, 139, 257, 467, 863, 1583, 2909, 5351, 9839, 18097, 33287, 61223, 112603, 207113, 380929, 700643, 1288657, 2370223, 4359517, 8018383, 14748119, 27126019, 49892519, 91766581, 168785119, 310444181

Offset: 0

Jonathan Vos Post, Mar 09 2007

Analog of A055500 a(0)=1, a(1)=1, a(n) = largest prime <= a(n-1)+a(n-2). Might be called the Prime-tribonacci sequence. a(n) is asymptotic to c*T^n where T is the tribonacci constant 1.83928675 whose digits are A058265 for a real constant c.

			a(3) = 3 = a(0)+a(1)+a(2) = 1+1+1 = 3.
a(4) = 5 = a(1)+a(2)+a(3) = 1+1+3 = 5.
a(5) = 7 < a(2)+a(3)+a(4) = 1+3+5 = 9.
a(6) = 13 < a(3)+a(4)+a(5) = 3+5+7 = 15.
a(7) = 23 < a(4)+a(5)+a(6) = 5+7+13 = 25.
a(8) = 43 = a(5)+a(6)+a(7) = 7+13+23 = 43.
a(9) = 79 = a(6)+a(7)+a(8) = 13+23+43 = 79.
a(10) = 139 < a(7)+a(8)+a(9) = 23+43+79 = 145.

Cf. A000040, A000073, A055500, A058265.

a[0]:=1:a[1]:=1:a[2]:=1:for n from 3 to 40 do a[n]:=prevprime(1+a[n-1]+a[n-2]+a[n-3]) od: seq(a[n],n=0..40); # Emeric Deutsch, Mar 24 2007

More terms from Emeric Deutsch, Mar 24 2007

Offset corrected by N. J. A. Sloane, Jun 16 2021

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A065435 a(3) = 2, a(4) = 3; for n > 4, a(n) = {a(n-2)}+{a(n-1)}, where {a} means largest prime <= a.

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A071353 First term of the continued fraction expansion of (3/2)^n.

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A345020 a(0) = a(1) = 1, a(n) = largest natural number m <= a(n-1) + a(n-2) where gcd(m,a(k)) = 1 for all 1 < k <= n-1.

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A126273 a(0) = a(1) = a(2) = 1, a(n) = largest prime <= a(n-1) + a(n-2) + a(n-3).

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