A011764 -id:A011764 - cp's OEIS frontend

A165421 a(1) = 1, a(2) = 3, a(n) = product of the previous terms for n >= 3.

1, 3, 3, 9, 81, 6561, 43046721, 1853020188851841, 3433683820292512484657849089281, 11790184577738583171520872861412518665678211592275841109096961

Offset: 1

Jaroslav Krizek, Sep 17 2009

nonn

Essentially a duplicate of A011764. - N. J. A. Sloane, Oct 06 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..14

a[1]:= 1; a[2]:= 3; a[n_]:= Product[a[j], {j,1,n-1}]; Table[a[n], {n, 1, 12}] (* G. C. Greubel, Oct 19 2018 *)
nxt[{prd_,a_}]:=Module[{c=prd*a},{c,prd*a}]; Join[{1,3},Rest[ NestList[ nxt,{1,3},10][[All,1]]]] (* Harvey P. Dale, Jan 31 2022 *)

{a(n) = if(n==1, 1, if(n==2, 3, prod(j=1,n-1, a(j))))};
for(n=1,10, print1(a(n), ", ")) \\ G. C. Greubel, Oct 19 2018

a(1) = 1, a(2) = 3, a(n) = Product_{i=1..n-1} a(i), n >= 3.
a(1) = 1, a(2) = 3, a(n) = A000244(2^(n-3)) = A011764(n-3) = 3^(2^(n-3)), n >= 3.
a(1) = 1, a(2) = 3, a(3) = 3, a(n) = (a(n-1))^2, n >= 4.

A059918 a(n) = (3^(2^n)-1)/2.

Original entry on oeis.org

1, 4, 40, 3280, 21523360, 926510094425920, 1716841910146256242328924544640, 5895092288869291585760436430706259332839105796137920554548480

Offset: 0

Henry Bottomley, Feb 08 2001

Denominator of b(n) where b(n) = 1/2*(b(n-1) + 1/b(n-1)), b(0)=2. - Vladeta Jovovic, Aug 15 2002

Harry J. Smith, Table of n, a(n) for n = 0..11

Cf. A059917 (numerators).
Cf. A059723, A059917, A059919, A011764.
Cf. A007089, A136308.

Array[(3^(2^#) - 1)/2 &, 8, 0] (* Michael De Vlieger, Feb 05 2022 *)

{ for (n=0, 11, write("b059918.txt", n, " ", (3^(2^n) - 1)/2); ) } \\ Harry J. Smith, Jun 30 2009

a(n) = a(n-1)*(3^(2^(n-1))+1) with a(0) = 1.
a(n) = (3^(2^n)-1)/2 = (A059723(n+1)-A059723(n))/A059723(n) = A059917(n)-1 = a(n-1)*A059919(n-1) = a(n-1)*(A011764(n-1)+1)
1 = Sum_{n>=0} 3^(2^n)/a(n+1). 1 = 3/4 + 9/40 + 81/3280 + 6561/21523360 + ...; with partial sums: 3/4, 39/40, 3279/3280, 21523359/21523360, ..., (a(n)-1)/a(n), ... . - Gary W. Adamson, Jun 22 2003
A136308(n) = A007089(a(n)). - Jason Kimberley, Dec 19 2012

A165427 a(1) = 1, a(2) = 9, a(n) = product of the previous terms for n >= 3.

Original entry on oeis.org

1, 9, 9, 81, 6561, 43046721, 1853020188851841, 3433683820292512484657849089281, 11790184577738583171520872861412518665678211592275841109096961

Offset: 1

Jaroslav Krizek, Sep 17 2009

G. C. Greubel, Table of n, a(n) for n = 1..13

a[1]:= 1; a[2]:= 9; a[n_]:= Product[a[j], {j,1,n-1}]; Table[a[n],{n,1, 12}] (* G. C. Greubel, Oct 19 2018 *)

{a(n) = if(n==1, 1, if(n==2, 9, prod(j=1,n-1, a(j))))};
for(n=1,10, print1(a(n), ", ")) \\ G. C. Greubel, Oct 19 2018

a(1) = 1, a(2) = 9, a(n) = Product_{i = 1..n-1} a(i), n >= 3.
a(1) = 1, a(2) = 9, a(n) = A001019(2^(n-3)) = 9^(2^(n-3)), n >= 3.
a(1) = 1, a(2) = 9, a(3) = 9, a(n) = (a(n-1))^2, n >= 4.
a(n) = A011764(n-2), n > 2. - R. J. Mathar, Sep 20 2009

A225156 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 3/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 2, 7, 67, 5623, 37772347, 1653794703916063, 3104205768420613437667191487267, 10767416908549848056705041797805600349527548164015760674541223

Offset: 1

Martin Renner, Apr 30 2013

Numerators of the sequence of fractions f(n) is A165421(n+1), hence sum(A165421(i+1)/a(i),i=1..n) = product(A165421(i+1)/a(i),i=1..n) = A165421(n+2)/A225163(n) = A011764(n-1)/A225163(n).

			f(n) = 3, 3/2, 9/7, 81/67, ...
3 + 3/2 = 3 * 3/2 = 9/2; 3 + 3/2 + 9/7 = 3 * 3/2 * 9/7 = 81/14; ...

Paul Yiu, Recreational Mathematics, Department of Mathematics, Florida Atlantic University, 2003, Chapter 5.4, p. 207 (Project).

Cf. A011764, A100441, A165421, A225163.

b:=n->3^(2^(n-2)); # n > 1
b(1):=3;
p:=proc(n) option remember; p(n-1)*a(n-1); end;
p(1):=1;
a:=proc(n) option remember; b(n)-p(n); end;
a(1):=1;
seq(a(i),i=1..9);

a(n) = 3^(2^(n-2)) - product(a(i),i=1..n-1), n > 1 and a(1) = 1.

a(n) = 3^(2^(n-2)) - p(n) with a(1) = 1 and p(n) = p(n-1)*a(n-1) with p(1) = 1.

A225161 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 9/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 8, 73, 5977, 39556153, 1714946746986937, 3196895220321005409761642330233, 11033196234263169646028268239301916905952651329069957632398777

Offset: 1

Martin Renner, Apr 30 2013

nonn

Numerators of the sequence of fractions f(n) is A165427(n+1), hence sum(A165427(i+1)/a(i),i=1..n) = product(A165427(i+1)/a(i),i=1..n) = A165427(n+2)/A225168(n) = A165421(n+3)/A225168(n) = A011764(n)/A225168(n).

			f(n) = 9, 9/8, 81/73, 6561/5977, ...
9 + 9/8 = 9 * 9/8 = 81/8; 9 + 9/8 + 81/73 = 9 * 9/8 * 81/73 = 6561/584; ...

Cf. A011764, A100441, A165421, A165427, A225168.

b:=n->9^(2^(n-2)); # n > 1
b(1):=9;
p:=proc(n) option remember; p(n-1)*a(n-1); end;
p(1):=1;
a:=proc(n) option remember; b(n)-p(n); end;
a(1):=1;
seq(a(i),i=1..8);

a(n) = 9^(2^(n-2)) - product(a(i),i=1..n-1), n > 1 and a(1) = 1.

a(n) = 9^(2^(n-2)) - p(n) with a(1) = 1 and p(n) = p(n-1)*a(n-1) with p(1) = 1.

A225163 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 3/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 2, 14, 938, 5274374, 199225484935778, 329478051871899046990657602014, 1022767669188735114815831063606918316150663428260080434555738

Offset: 1

Martin Renner, Apr 30 2013

nonn

Numerators of the sequence s(n) of the sum resp. product of fractions f(n) is A165421(n+2), hence s(n) = sum(A165421(i+1)/A225156(i),i=1..n) = product(A165421(i+1)/A225156(i),i=1..n) = A165421(n+2)/a(n) = A011764(n-1)/a(n).

			f(n) = 3, 3/2, 9/7, 81/67, ...
3 + 3/2 = 3 * 3/2 = 9/2; 3 + 3/2 + 9/7 = 3 * 3/2 * 9/7 = 81/14; ...
s(n) = 1/b(n) = 3, 9/2, 81/14, ...

Paul Yiu, Recreational Mathematics, Department of Mathematics, Florida Atlantic University, 2003, Chapter 5.4, p. 207 (Project).

Cf. A011764, A076628, A165421, A225156.

b:=proc(n) option remember; b(n-1)-b(n-1)^2; end:
b(1):=1/3;
a:=n->3^(2^(n-1))*b(n);
seq(a(i),i=1..9);

a(n) = 3^(2^(n-1))*b(n) where b(n)=b(n-1)-b(n-1)^2 with b(1)=1/3.

A241241 If x is in the sequence then so are x^2 and x(x+1)/2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 10, 16, 21, 36, 45, 55, 81, 100, 136, 231, 256, 441, 666, 1035, 1296, 1540, 2025, 3025, 3321, 5050, 6561, 9316, 10000, 18496, 26796, 32896, 53361, 65536, 97461, 194481, 222111, 443556, 536130, 840456, 1071225, 1186570, 1679616, 2051325

Offset: 1

Reinhard Zumkeller, Apr 17 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000290, A000217.

Subsequence of A005214; some subsequences: A001146, A007501, A011764, A176594, A173501, A050909.

import Data.Set (singleton, deleteFindMin, insert)
a241241 n = a241241_list !! (n-1)
a241241_list = 0 : 1 : f (singleton 2) where
   f s = m : f (insert (a000290 m) $ insert (a000217 m) s')
         where (m, s') = deleteFindMin s

Nest[Flatten[{#,#^2,(#(#+1))/2}]&,{0,1,2},5]//Union (* Harvey P. Dale, Aug 12 2016 *)

Initial 0 and 1 prepended by Jon Perry, Apr 17 2014

A079271 a(n) = 4 * a(n-1) * (3^(2^(n-1))-a(n-1)) with a(0)=1.

Original entry on oeis.org

1, 8, 32, 6272, 7250432, 1038154236987392, 3383826162019367796397224108032, 674838593766753484487654913831820720085359667709963001167872

Offset: 0

Henry Bottomley, Feb 06 2003

a(n) is the numerator of b(n)=a(n)/3^(2^n)=a(n)/A011764(n) which is a logistic chaotic sequence of reals in (0,1) with b(n)=4*b(n-1)*(1-b(n-1)) starting at b(0)=1/3; the truncated values of b(n) start: 0.333..., 0.888..., 0.395..., 0.955..., 0.168..., 0.560..., 0.985..., 0.057..., 0.215..., etc.

A225168 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 9/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 8, 584, 3490568, 138073441864904, 236788599971507074896206759048, 756988343475413525492604622110601759725560263205883476698184

Offset: 1

Martin Renner, Apr 30 2013

nonn

Numerators of the sequence s(n) of the sum resp. product of fractions f(n) is A165427(n+2), hence sum(A165427(i+1)/A225161(i),i=1..n) = product(A165427(i+1)/A225161(i),i=1..n) = A165427(n+2)/a(n) = A165421(n+3)/a(n) = A011764(n)/a(n).

			f(n) = 9, 9/8, 81/73, 6561/5977, ...
9 + 9/8 = 9 * 9/8 = 81/8; 9 + 9/8 + 81/73 = 9 * 9/8 * 81/73 = 6561/584; ...
s(n) = 1/b(n) = 9, 81/8, 6561/584, ...

Cf. A011764, A076628, A165421, A165427, A225161.

b:=proc(n) option remember; b(n-1)-b(n-1)^2; end:
b(1):=1/9;
a:=n->9^(2^(n-1))*b(n);
seq(a(i),i=1..8);

a(n) = 9^(2^(n-1))*b(n) where b(n)=b(n-1)-b(n-1)^2 with b(1)=1/9.

A361253 If n = m^2 for some m > 1 then a(n) = a(m), otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 8, 65, 66, 67, 68

Offset: 0

Rémy Sigrist, Mar 06 2023

All terms belong to A000037 U { 0, 1 }.
All terms of A000037 appear infinitely many times.
This sequence can be seen as the limit of the k-th iterate of A097448 as k tends to infinity.

			a(9) = a(3^2) = a(3) = 3 (as 3 is not a square).

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000037, A001146, A011764, A176594, A097448.

nn = 120; Array[Set[a[#], #] &, 2, 0]; Do[If[IntegerQ[#], Set[k, a[#]], Set[k, n]] &[Sqrt[n]]; Set[a[n], k], {n, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 06 2023 *)

a(n) = my (m); { while (n > 1 && issquare(n, &m), n = m); return (n) }

from sympy import integer_nthroot
def A361253(n):
    if n <= 1:
        return n
    a, b = integer_nthroot(c:=n,2)
    while b:
        a, b = integer_nthroot(c:=a,2)
    return c # Chai Wah Wu, Mar 17 2023

a(a(n)) = a(n).
a(n) <= A097448(n).
a(n) = 2 iff n belongs to A001146.
a(n) = 3 iff n belongs to A011764.
a(n) = 5 iff n belongs to A176594.

cp's OEIS Frontend

A165421 a(1) = 1, a(2) = 3, a(n) = product of the previous terms for n >= 3.

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A059918 a(n) = (3^(2^n)-1)/2.

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A165427 a(1) = 1, a(2) = 9, a(n) = product of the previous terms for n >= 3.

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A225156 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 3/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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A225161 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 9/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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A225163 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 3/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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A241241 If x is in the sequence then so are x^2 and x(x+1)/2.

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A079271 a(n) = 4 * a(n-1) * (3^(2^(n-1))-a(n-1)) with a(0)=1.

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A225168 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 9/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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