A005185 -id:A005185 - cp's OEIS frontend

A284478 a(n) = A005185(n) - A055748(n).

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 0, 2, 1, 2, 3, 3, 3, 2, 2, 2, 5, 1, -1, 1, 2, 1, 0, 4, 1, 1, 4, 5, 3, 3, 4, 3, 4, 5, 5, 5, 3, 1, 3, 4, 8, -1, -1, 2, 1, 0, 0, 0, -1, 2, 0, 2, 1, 0, 0, 8, 1, -1, 6, 3, 1, 7, 7, 3, 4, 6, 5, 6, 5, 8, 5, 4, 5, 7, 7, 4, 8, 3, 7, 6, 4, 7, 9, 2, 0, 3, 2, 2, 18

Offset: 1

Altug Alkan, Mar 27 2017

			a(4) = 1 because a(4) = A005185(4) - A055748(4) = 3 - 2 = 1.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Scatterplot of A284478

Cf. A005185, A055748.

A005185:= proc(n) option remember; procname(n-procname(n-1)) +procname(n-procname(n-2)) end proc:
A005185(1):= 1: A005185(2):= 1:
A055748:= proc(n) option remember; procname(procname(n-1)) +procname(n-procname(n-2)-1) end proc:
A055748(1):= 1: A055748(2):= 1:
A284478:= map(A005185 - A055748, [$1..1000]):
A284478:= seq(A284478[i], i=1..1000); # Altug Alkan, Apr 01 2017

a[n_] := a[n] = If[n < 3, 1, a[n - a[n - 1]] + a[n - a[n - 2]]]; b[n_] := b[n] = If[n < 3, 1, b[b[n - 1]] + b[n - b[n - 2] - 1]]; Array[a[#] - b[#] &, 96] (* Michael De Vlieger, Mar 29 2017 *)

q=vector(1000); c=vector(1000); q[1]=q[2]=1; for(n=3, #q, q[n]=(q[n-q[n-1]]+q[n-q[n-2]])); c[1]=c[2]=1; for(n=3, #c, c[n]=(c[c[n-1]]+c[n-c[n-2]-1])); va = vector(1000, n, q[n]-c[n])

A317161 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n)) where t = A005185. a(n) = 2*b(n) - n.

Original entry on oeis.org

1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 2, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0, 1, -2, -1, 0, -1, -2, -3, -2, -1, 0, 1, 0, -1, 0, -1, -2, 1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 2, 1, 0, 3, 0, 1, 2, 1, 2, 3, 2, 1, 2, 3, 0, 3, 2, 3, 4, 1, 2, 1, 0, -1, 0, 1, 0, 3, 0, -1

Offset: 1

Altug Alkan, Aug 07 2018

If there is a limiting value l such that lim_{n->infinity} b(n)/n = lim_{n->infinity} t(n)/n = l, then l = 1/2. So this sequence has definition a(n) = 2*b(n) - n.

Altug Alkan, Table of n, a(n) for n = 1..24064
Altug Alkan, Scatterplot of a(n) for n <= 24064

Cf. A005185, A317754.

Block[{t = NestWhile[Function[{a, n}, Append[a, a[[n - a[[-1]] ]] + a[[n - a[[-2]] ]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, Last@ # < 65 &], b}, b = NestWhile[Function[{b, n}, Append[b, n - b[[t[[n]] ]] - b[[n - t[[n]] ]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, Length@ # < Length@ t &]; Array[2 b[[#]] - # &, Length@ b] ] (* Michael De Vlieger, Aug 08 2018 *)

t=vector(99); t[1]=t[2]=1; for(n=3, #t, t[n] = t[n-t[n-1]]+t[n-t[n-2]]); b=vector(99); b[1]=b[2]=1; for(n=3, #b, b[n] = n-b[t[n]]-b[n-t[n]]); vector(99, k, 2*b[k]-k)

A327284 Indices of terms of the Hofstadter Q-sequence (A005185) not used in generating a later term.

Original entry on oeis.org

1, 216, 361, 429, 451, 599, 766, 774, 775, 778, 792, 793, 820, 849, 863, 882, 968, 1042, 1111, 1215, 1216, 1228, 1407, 1524, 1528, 1542, 1543, 1550, 1551, 1573, 1653, 1672, 1673, 1674, 1675, 1863, 1905, 1920, 1938, 1954, 2078, 2185, 2186, 2187, 2195, 2196, 2277

Offset: 1

Benjamin Chaffin, Sep 15 2019

nonn

Each term of the Q-sequence is the sum of two previous terms. This sequence gives the Q-sequence terms which appear never to be used as one of those two. In other words, a(n) != x - A005185(x-1), and a(n) != x - A005185(x-2).

Benjamin Chaffin, Table of n, a(n) for n = 1..10000

Cf. A005185, A076701.

A362257 a(n) = 2*Q(n) - n, where Q(n) is Hofstadter's Q-sequence A005185.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 3, 2, 3, 2, 1, 4, 3, 2, 5, 2, 3, 4, 3, 4, 3, 2, 1, 8, 3, 2, 5, 4, 3, 2, 9, 2, 1, 6, 7, 2, 3, 6, 3, 4, 5, 4, 5, 4, 3, 2, 1, 16, -1, 0, 9, 4, -1, 6, 5, 0, 7, 2, 5, 4, 3, 2, 17, 2, -3, 10, 3, -2, 9, 10, 3, 4, 7, 4, 5, 2, 7, 2, 3, 6, 7, 4, 3, 8

Offset: 1

Nathan Fox and Alexis Ducote, Apr 13 2023

sign

Just as for A005185, it is not known if this sequence exists for all n.

A005185 and this sequence exist as long |a(n)| remains less than n.

Cf. A005185, A239913.

from functools import lru_cache
@lru_cache(maxsize=None)
def A362257(n): return 2-n if n <= 2 else ((a:=1-A362257(n-1))+(b:=2-A362257(n-2))>>1)+A362257(n+a>>1)+A362257(n+b>>1) # Chai Wah Wu, Apr 13 2023

a(1) = 1, a(2) = 0; a(n) = (1/2)*(3 - a(n-1) - a(n-2)) + a((1/2)*(n + 1 - a(n-1))) + a((1/2)*(n + 2 - a(n-2))) for n >= 3.

A076267 Numbers n such that A005185(n) divides n.

Original entry on oeis.org

1, 2, 50, 56, 128, 156, 166, 208, 238, 272, 308, 336, 392, 474, 476, 512, 618, 658, 666, 710, 734, 836, 868, 1016, 1064, 1376, 1386, 1424, 1432, 1832, 2216, 2280, 2334, 2606, 2638, 2676, 2700, 2740, 2782, 2786, 2912, 2922, 2948, 2954, 3758, 4260, 4632

Offset: 1

Benoit Cloitre, Nov 05 2002

nonn

Is a(n)/n^2 bounded ?

Superset of A027619. [R. J. Mathar, Sep 21 2008]

			The 56th term of the Hofstadter Q-sequence is 28, which divides 56, hence 56 is in the sequence.

Cf. A005185.

A087720 Repeated terms in A005185.

Original entry on oeis.org

1, 3, 5, 6, 8, 11, 12, 14, 16, 17, 23, 24, 30, 32, 43, 47, 48, 54, 60, 72, 80, 83, 84, 92, 94, 96, 148, 172, 175, 178, 185, 186, 192, 215, 222, 228, 230, 252, 273, 301, 308, 338, 350, 372, 380, 382, 384, 408, 468, 517, 574, 593, 624, 673, 678, 738, 748, 762, 764, 789

Offset: 1

Roger L. Bagula, Sep 29 2003

nonn

Cf. A005185.

digits=30000;
Hofstadter[n_Integer?Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-2]];
Hofstadter[1] = Hofstadter[2] = 1;
a1=Table[Hofstadter[n], {n, 1, digits}];
b=Table[If[a1[[n]]-a1[[n-1]]==0, a1[[n]], 0], {n, 2, digits}];
c=Delete[Union[b], 1]

A087721 Strictly increasing domain of A005185.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 10, 11, 12, 16, 20, 21, 22, 23, 24, 25, 30, 32, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 52, 54, 56, 58, 60, 61, 62, 64, 66, 68, 71, 72, 73, 77, 78, 79, 80, 82, 83, 85, 87, 88, 90, 91, 92, 93, 94, 96, 101, 106, 108, 109, 111, 114, 115, 118, 120, 122, 123

Offset: 1

Roger L. Bagula, Sep 29 2003

nonn

Cf. A005185, A087722, A087723.

digits=750;
Hofstadter[n_Integer?Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-2]];
Hofstadter[1] = Hofstadter[2] = 1;
a1=Table[Hofstadter[n], {n, digits}];
f[x_, y_] := x-y/; x-y>0;
f[x_, y_] := 0/; x-y<=0;
b=Table[If[f[a1[[n]], a1[[n-1]]]>0, a1[[n]], 0], {n, 2, digits}];
c=Delete[Union[b], 1]

A087742 a(n) = 1+Abs[Prime[A005185[n]]-A005185[Prime[n]]].

Original entry on oeis.org

2, 1, 1, 1, 2, 2, 2, 1, 2, 4, 8, 2, 5, 6, 6, 4, 4, 2, 5, 1, 4, 5, 7, 8, 3, 6, 2, 2, 8, 2, 4, 5, 13, 7, 8, 9, 13, 6, 15, 9, 12, 11, 8, 18, 6, 22, 26, 2, 24, 24, 8, 14, 32, 17, 31, 29, 21, 23, 13, 21, 18, 6, 12, 29, 28, 4, 23, 39, 11, 3, 21, 17, 14, 24, 20, 26, 20, 57, 10, 20, 23, 28, 40, 36, 30

Offset: 1

Roger L. Bagula, Oct 01 2003

nonn

A "commutator" between the Hofstadter A005185 sequence and the primes.

Cf. A000040, A005185.

Hofstadter[n_Integer?Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-2]] Hofstadter[1] = Hofstadter[2] = 1 digits=200 a=Table[1+Abs[Prime[Hofstadter[n]]-Hofstadter[Prime[n]]], {n, 1, digits}]

Edited by N. J. A. Sloane, Nov 08 2005

A087753 a(n) = least k such that A005185(k)=n; or 0 if there is no such k.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 0, 12, 16, 15, 18, 20, 0, 25, 0, 24, 32, 0, 36, 31, 35, 38, 41, 43, 50, 53, 0, 52, 0, 51, 65, 48, 64, 0, 67, 0, 71, 66, 69, 63, 79, 77, 80, 81, 86, 84, 87, 90, 0, 104, 0, 98, 110, 99, 116, 100, 125, 108, 0, 105, 123, 114, 129, 96, 139, 115, 0, 118, 142, 0, 126

Offset: 1

Roger L. Bagula, Oct 02 2003

nonn

Cf. A005185.

Edited by N. J. A. Sloane, Nov 08 2005

A088492 a(2n+1)=2n+1, a(2n) = floor(2*n/A005185(n)), a weighted inverse of Hofstadter's Q-sequence.

Original entry on oeis.org

2, 3, 4, 5, 3, 7, 2, 9, 3, 11, 3, 13, 2, 15, 3, 17, 3, 19, 3, 21, 3, 23, 3, 25, 3, 27, 3, 29, 3, 31, 3, 33, 3, 35, 3, 37, 3, 39, 3, 41, 3, 43, 3, 45, 3, 47, 3, 49, 3, 51, 3, 53, 3, 55, 3, 57, 3, 59, 3, 61, 3, 63, 3, 65, 3, 67, 3, 69, 3, 71, 3, 73, 3, 75, 3, 77, 3, 79, 3, 81, 3, 83, 3, 85, 3, 87

Offset: 2

Roger L. Bagula, Nov 10 2003

nonn

Define a sequence of partial products of Hofstadters Q-sequence, H(n) = prod_{i=1..n} A005185(i) = 1, 1, 2, 6, 18, 72, 360, 1800,.. for n>=1 (which differs from A004395).
The original definition was equivalent to a(n) = [n *H([(n-1)/2])/H([n/2])] where [..] is the floor function.
A consequence of this construction is a(2n+1)=2n+1 at the odd indices. At the even indices, a(2*n) = [2*n*H(n-1)/H(n)] = [2*n/A005185(n)], which is used to simplify the definition.
a(2n)=3 for 8<=n<=48. The first 5 at an even index occurs at a(2*193)=5.

Cf. A005185.

Hofstadter[n_Integer?Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-2]] Hofstadter[1] = Hofstadter[2] = 1 p[n_]=n!/Product[Hofstadter[i], {i, 1, Floor[n/2]}] digits=200 a0=Table[Floor[p[n]/p[n-1]], {n, 2, digits}]

Definition replaced and comment added - R. J. Mathar, Dec 08 2010

cp's OEIS Frontend

A284478 a(n) = A005185(n) - A055748(n).

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A317161 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n)) where t = A005185. a(n) = 2*b(n) - n.

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A327284 Indices of terms of the Hofstadter Q-sequence (A005185) not used in generating a later term.

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A362257 a(n) = 2*Q(n) - n, where Q(n) is Hofstadter's Q-sequence A005185.

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A076267 Numbers n such that A005185(n) divides n.

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A087720 Repeated terms in A005185.

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A087721 Strictly increasing domain of A005185.

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A087742 a(n) = 1+Abs[Prime[A005185[n]]-A005185[Prime[n]]].

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A087753 a(n) = least k such that A005185(k)=n; or 0 if there is no such k.

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A088492 a(2n+1)=2n+1, a(2n) = floor(2*n/A005185(n)), a weighted inverse of Hofstadter's Q-sequence.

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