A130535 -id:A130535 - cp's OEIS frontend

A283207 a(n) = a(floor(n/a(n-1))) + a(floor(n/a(n-2))) with a(1) = a(2) = 2.

2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 6, 4, 6, 6, 4, 8, 6, 6, 8, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Altug Alkan, Mar 03 2017

nonn

For the first 10^6 terms, the maximum value of a(n) is 64 and the values of b(n) = least k such that a(k) = 2*n are 1, 3, 13, 12, 105, 97, 126, 96, 1681, 1552, 1746, 1537, 1734, 1926, 4050, 1536, 53793, 49665, 53890, 49185, 53862, 57024, 55616, 49153, 55488, 55302, 81249, 61446, 83619, 115214, 162000, 49152; note that b(2^n) = 3*2^((n+2)*(n-1)/2) for n = 1 to 5.

This sequence is a_{1,2}(n) where a_{r,s}(n) = a_{r,s}(floor(n/a_{r,s}(n-r))) + a_{r,s}(floor(n/a_{r,s}(n-s))) with a_{r,s}(n) = 2 for n <= s (r < s). - Altug Alkan, Jun 28 2020

			a(5) = 4 because a(5) = a(floor(5/a(4))) + a(floor(5/a(3))) = a(floor(5/4)) + a(floor(5/4)) = a(1) + a(1) = 4.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Graph of A283207
Altug Alkan, Line plot of (a(floor(n/a(n-1))), a(floor(n/a(n-2)))) for n <= 2^15
Rémy Sigrist, Scatterplot of the first 2^34 terms
Rémy Sigrist, Line plot of (a(floor(n/a(n-1))), a(floor(n/a(n-2)))) for n <= 12855108032

Cf. A005185, A130535.

A:= Vector(100):
A[1]:= 2: A[2]:= 2:
for n from 3 to 100 do A[n]:= A[floor(n/A[n-1])] + A[floor(n/A[n-2])] od:
convert(A,list); # Robert Israel, Jun 23 2020

a[1] = a[2] = 2; a[n_] := a[n] = a[Floor[n/a[n - 1]]] + a[Floor[n/a[n - 2]]]; Array[a, 120] (* Michael De Vlieger, Mar 06 2017 *)

a=vector(100); a[1]=a[2]=2; for(n=3, #a, a[n]=a[n\a[n-1]]+a[n\a[n-2]]); a

from functools import lru_cache
@lru_cache(maxsize=None)
def A283207(n):
    return 2 if n <= 2 else A283207(n//A283207(n-1)) + A283207(n//A283207(n-2)) # Chai Wah Wu, Jun 23 2020

A335898 a(n) = a(floor((n-1)/a(n-1))) + a(floor((n-2)/a(n-2))) with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 2, 3, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 4, 5, 6, 5, 5, 6, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 4, 6, 6, 5, 6, 6, 6, 6, 6, 7, 6, 6, 8, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 7, 7, 6, 7

Offset: 1

Altug Alkan, following a suggestion from Andrew R. Booker, Jun 29 2020

This sequence is a_1(n) where a_i(n) = Sum_{k=1..i+1} a_i(floor((n-k)/a_i(n-k))) with a_i(n) = 1 for n <= i+1.

Conjecture: This sequence hits every positive integer.

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A130535, A283207.

a[1] = a[2] = 1; a[n_] := a[n] = a[Floor[(n-1)/a[n-1]]] + a[Floor[(n-2)/a[n-2]]]; Array[a, 100] (* Amiram Eldar, Jun 29 2020 *)

a=vector(10^2); a[1]=a[2]=1; for(n=3, #a, a[n]=a[(n-1)\a[n-1]]+a[(n-2)\a[n-2]]); a

A335901 a(n) = 2*a(floor((n-1)/a(n-1))) with a(1) = 1.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 4, 8, 4, 4, 4, 4, 4, 8, 4, 8, 4, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 4, 8, 4, 8, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 8, 4, 8, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Altug Alkan, following a suggestion from Andrew R. Booker, Jun 29 2020

Least k such that a(k) = 2^n are 1, 2, 5, 21, 169, 2705, ... (Conjecture: This sequence is A117261).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A130147, A132424, A130535, A283207, A288914, A335898.

f:= proc(n) option remember;
  2*procname(floor((n-1)/procname(n-1))) end proc:
f(1):= 1:
map(f, [$1..105]); # Robert Israel, Jul 08 2020

a[1] = 1; a[n_] := a[n] = 2 * a[Floor[(n-1)/a[n-1]]]; Array[a, 100] (* Amiram Eldar, Jun 29 2020 *)

a=vector(10^3); a[1]=1; for(n=2, #a, a[n]=2*a[(n-1)\a[n-1]]); a

cp's OEIS Frontend

A283207 a(n) = a(floor(n/a(n-1))) + a(floor(n/a(n-2))) with a(1) = a(2) = 2.

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A335898 a(n) = a(floor((n-1)/a(n-1))) + a(floor((n-2)/a(n-2))) with a(1) = a(2) = 1.

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A335901 a(n) = 2*a(floor((n-1)/a(n-1))) with a(1) = 1.

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Mathematica

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