A004001 -id:A004001 - cp's OEIS frontend

A143232 Sum of denominators of Egyptian fraction expansion of A004001(n) - n/2.

2, 0, 2, 0, 2, 1, 2, 0, 2, 1, 3, 1, 3, 1, 2, 0, 2, 1, 3, 2, 3, 2, 4, 2, 4, 2, 3, 2, 3, 1, 2, 0, 2, 1, 3, 2, 4, 2, 4, 3, 5, 3, 5, 4, 5, 4, 5, 3, 5, 4, 5, 4, 5, 3, 5, 3, 4, 2, 4, 2, 3, 1, 2, 0, 2, 1, 3, 2, 4, 3, 4, 3, 5, 4, 6, 4, 6, 5, 7, 5, 7, 6, 7, 6, 7, 5, 7, 6, 8, 6, 8, 7, 8, 7, 8, 6, 8, 7, 8, 7, 8, 6, 8, 6, 7

Offset: 1

Dan Bron (j (at) bron.us), Jul 31 2008

A004001 is the Hofstadter-Conway $10,000 Sequence and A004001(n) - n/2 is increasingly larger versions of the batrachion Blancmange function.

			a(43) = 5 because A004001(43) = 25, so (A004001(43) - (43/2)) = 3.5 and the Egyptian fraction expansion of 3.5 is (1/1)+(1/1)+(1/1)+(1/2), so the denominators are 1,1,1,2 which sums to 5.

G. C. Greubel, Table of n, a(n) for n = 1..10000
H. Rich, Hofstadter-Conway $10,000 Sequence, Programming Archives, 2008.
Eric Weisstein's World of Mathematics, Hofstadter-Conway $10,000 Sequence.

Cf. A004001.

a004001 =: ((] +&:$: -) $:@:<:)@.(2&<) M.
a143232 =: (+ 3 * 1&|)@:(a004001 - -:)"0

A004001:=[n le 2 select 1 else Self(Self(n-1))+ Self(n-Self(n-1)):n in [1..125]];
f:= func< x | 4*x - 3*Floor(x) >;
A143232:= func< n | f(A004001[n] - n/2) >;
[A143232(n): n in [1..100]]; // G. C. Greubel, Sep 10 2024

HC[n_]:= HC[n]= If[n<3, 1, HC[HC[n-1]] +HC[n-HC[n-1]]]; (*HC=A004001*)
f[x_]:= 4*x -3*Floor[x];
A143232[n_]:= f[HC[n] -n/2];
Table[A143232[n], {n,100}] (* G. C. Greubel, Sep 10 2024 *)

@CachedFunction
def b(n): # b = A004001
    if n<3: return 1
    else: return b(b(n-1)) + b(n-b(n-1))
def f(x): return x + 3 * (x - floor(x))
def A143232(n): return f(b(n) - n/2)
[A143232(n) for n in range(1,101)] # G. C. Greubel, Sep 10 2024

a(n) = Sum of denominators of Egyptian fraction expansion of A004001(n) - n/2 .

For practical purposes, a full Egyptian fraction algorithm is not necessary. Since the elements of A004001(n) - n/2 are either whole or their fractional part is .5, the sequence can be effected by a(n) = sefd(A004001(n) - n/2) with sefd(x) = x + 3 * (x - floor(x)) .

A147981 a(n) = A147952(A004001(n)).

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Nov 18 2008

nonn

Cf. A004001, A147952.

(*A004001*) g[0] = 0; g[1] = 1; g[2] = 1; g[n_] := g[n] = g[g[n - 1]] + g[n - g[n - 1]]; (*A147952*) f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 2]] + If[Mod[n, 3] == 0, f[f[n/3]], If[Mod[n, 3] ==1, f[f[(n - 1)/3]], f[n - f[(n - 2)/3]]]]; Table[f[g[n]], {n, 0, 100}]

a(n) = A147952(A004001(n)) for n >= 0 with A004001(0) := 0.

Name edited by Petros Hadjicostas, Apr 22 2020

A154280 List of pairs (a(n),b(n)): f(n) = A004001(n); a(n) = f(n) + a(n-1); b(n) = f(n)*b(n-1).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 4, 2, 6, 4, 9, 12, 13, 48, 17, 192, 21, 768, 26, 3840, 32, 23040, 39, 161280, 46, 1128960, 54, 9031680, 62, 72253440, 70, 578027520, 78, 4624220160, 87, 41617981440, 97, 416179814400, 108, 4577977958400, 120, 54935735500800, 132

Offset: 0

Roger L. Bagula, Jan 06 2009

G. C. Greubel, Table of n, a(n) for n = 0..1001

Cf. A004001.

(*A004001*) f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]];
a[0] = 0; a[n_] := a[n] = f[n] + a[n - 1];
b[0] = 1; b[1] = 1; b[n_] := b[n] = (f[n])*b[n - 1];
Flatten[Table[{a[n], b[n]}, {n, 0, 30}]]

A176047 Triangle t(n,m) = A004001(m+1)+A004001(n-m+1)-A004001(n+1) read by rows, 0<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, -1, 0, 1, 0, -1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 0

Roger L. Bagula and Gary W. Adamson, Apr 07 2010

			1;
1, 1;
1, 0, 1;
1, 1, 1, 1;
1, 0, 1, 0, 1;
1, 0, 0, 0, 0, 1;
1, 1, 1, 0, 1, 1, 1;

A176047 := proc(n,m)
        A004001(m+1)+A004001(n-m+1)-A004001(n+1) ;
end proc:
seq(seq(A176047(n,m),m=0..n),n=0..12) ; # R. J. Mathar, Jul 11 2012

Mutually compatible definition and terms installed. Erroneous row sums removed. - R. J. Mathar, Jul 11 2012

A283525 Remainder when sum of first n terms of A004001 is divided by 3*n.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 17, 21, 26, 2, 6, 10, 15, 20, 25, 30, 36, 43, 51, 0, 6, 13, 21, 29, 38, 47, 56, 66, 76, 86, 3, 10, 18, 27, 37, 48, 60, 72, 85, 99, 114, 3, 16, 30, 44, 59, 74, 89, 105, 122, 139, 1, 16, 31, 47, 63, 79, 95, 112, 129, 146, 163, 180, 5, 20, 36, 53, 71, 90, 110, 130, 151, 173, 196, 220, 16, 38, 61, 85, 109

Offset: 1

Altug Alkan, Mar 10 2017

nonn

Sequence represents b(n, 3) where b(n, i) = (Sum_{k=1..n} A004001(k)) mod (n*i). See also A282891, A283501 and corresponding illustrations in Links section.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Illustration of Residue Classes Modulo 12
Altug Alkan, Alternative Scatterplot of A283525

Cf. A004001, A282891, A283501.

A004001:= proc(n) option remember; procname(procname(n-1)) +procname(n-procname(n-1)) end proc:
A004001(1):= 1: A004001(2):= 1:
L:= ListTools[PartialSums](map(A004001, [$1..1000])):
seq(L[i] mod (3*i), i=1..1000); # after Robert Israel at A282891

b[1] = 1; b[2] = 1; b[n_] := b[n] = b[b[n - 1]] + b[n - b[n - 1]]; a[n_] := Mod[Sum[b[k], {k, n}], 3 n]; Array[a, 80] (* Robert G. Wilson v, Mar 13 2017 *)

a=vector(1000); a[1]=a[2]=1; for(n=3, #a, a[n]=a[a[n-1]]+a[n-a[n-1]]); vector(#a, n, sum(k=1, n, a[k]) % (3*n))

a(n) = (Sum_{k=1..n} A004001(k)) mod (3*n).

A317921 a(1) = a(2) = 1; for n >= 3, a(n) = 3*a(t(n)) - a(n-t(n)) where t = A004001.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 4, 4, 13, 13, 10, 7, 7, 7, 8, 8, 35, 35, 26, 17, 8, 8, 8, 8, 11, 14, 17, 17, 17, 17, 16, 16, 97, 97, 70, 43, 16, -11, -11, -11, -11, -11, -2, 7, 16, 25, 34, 43, 43, 43, 43, 43, 43, 43, 40, 37, 34, 31, 31, 31, 31, 31, 32, 32, 275, 275, 194, 113, 32, -49, -130, -130, -130, -130, -130

Offset: 1

Altug Alkan, Aug 11 2018

Sequence has a fractal-like structure. Each generation (between consecutive powers of 2) provides a pattern which looks like an EKG signal since maximum value of a(n) (in corresponding generation) is damped step by step.

Altug Alkan, Table of n, a(n) for n = 1..32768
Altug Alkan, Line plot of a(n) for n <= 2^17

Cf. A004001, A317648, A317754.

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

A051284 a(n) is the number k, 2^n < k < 2^(n+1), such that k/c(k) is a minimum in the interval, where c(k) is Hofstadter-Conway sequence A004001.

Original entry on oeis.org

3, 6, 11, 23, 44, 92, 178, 370, 719, 1487, 2897, 5969, 11651, 22223, 45083, 89516, 181385, 353683, 722589, 1423078, 2903564, 5696576, 11635316, 22866150, 46704206, 91835554, 187298550

Offset: 1

Jud McCranie

The ratio of k/c(k) (where c(k)=A004001) reaches a maximum of 2.0 when n is a power of 2. When n=6 the ratio has a relative minimum of 1.5, so a(2) = 6.

Cf. A004001.

a(26)-a(27) and title clarified by Sean A. Irvine, Sep 01 2021

A087741 a(n) = 1+Abs[A000040[A004001[n]]-A004001[A000040[n]]].

Original entry on oeis.org

2, 1, 1, 2, 3, 2, 3, 5, 4, 4, 2, 5, 6, 7, 9, 12, 10, 4, 5, 2, 4, 5, 5, 9, 9, 11, 12, 7, 8, 10, 12, 15, 14, 14, 16, 13, 14, 19, 16, 15, 13, 14, 10, 7, 10, 9, 14, 19, 15, 15, 16, 14, 14, 16, 3, 9, 14, 16, 17, 20, 21, 27, 39, 41, 36, 37, 36, 38, 36, 30, 34, 34, 33, 30, 31, 32, 26, 28, 26, 29

Offset: 1

Roger L. Bagula, Oct 01 2003

A "commutator" between the sequence of primes A000040 and the Conway-Hofstadter A004001 sequence.

This sequence has a nice graph. ListPlot[a,PlotRange->All,PlotJoined->True]

Cf. A004001, A087740.

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

A089996 a(n) = primes generated by the function ( f[n_]=Floor[(A004001[n]Prime[n])Log[2]/(2*PrimePi[n]+1)]).

Original entry on oeis.org

3, 5, 13, 17, 41, 53, 59, 61, 101, 127, 151, 167, 193, 269, 277, 281, 283, 313, 359, 419, 421, 439, 463, 467, 499, 509, 619, 691, 743, 787, 853, 859, 907, 1061, 1069, 1097, 1181, 1229, 1249, 1277, 1289, 1303, 1381, 1427, 1453, 1531, 1571, 1583, 1609, 1741

Offset: 1

Roger L. Bagula, Jan 14 2004

nonn

A prime generating function based on the primes, A004001 and the distribution of the primes.

By itself the integer function : f[n_]=Floor[(Conway[n]*Prime[n])*Log[2]/(2*PrimePi[n]+1)] is not very interesting: it is made to match the function g[n_]=n*Log[n]

Cf. A004001.

digits=6*200 Conway[n_Integer?Positive] := Conway[n] =Conway[Conway[n-1]] + Conway[n - Conway[n-1]] Conway[1] = Conway[2] = 1 (* PrimeQ sieve function *) a=Table[If[PrimeQ[Floor[(Conway[n]*Prime[n])*Log[2]/(2*PrimePi[n]+1)]]==True, Floor[(Conway[n]*Prime[n])*Log[2]/(2*PrimePi[n]+1)], 0], {n, 1, digits}] (* eliminate the extra zeros *) b=Union[a] Delete[b, 1]

A094528 Triangle T(n,k) read by rows related to A004001.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 2, 3, 4, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 4, 6, 6, 6, 7, 5, 5, 4, 6, 7, 7, 7, 8, 6, 6, 5, 7, 8, 8, 8, 8, 9, 6, 6, 6, 7, 8, 9, 9, 9, 9, 10, 7, 6, 7, 7, 9, 10, 10, 10, 10, 10, 11, 7, 6, 8, 8, 10, 10, 11, 11, 11, 11, 11, 12, 7, 7, 8, 8, 10, 11, 12, 12, 12, 12, 12, 12, 13

Offset: 1

Benoit Cloitre, Jun 05 2004

T(n, k)=b(2^k+n)-2^(k-1) where b(n)=A004001(n) (n=1, 2, 3, ... 1<=k<=n)

cp's OEIS Frontend

A143232 Sum of denominators of Egyptian fraction expansion of A004001(n) - n/2.

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A147981 a(n) = A147952(A004001(n)).

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A154280 List of pairs (a(n),b(n)): f(n) = A004001(n); a(n) = f(n) + a(n-1); b(n) = f(n)*b(n-1).

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A176047 Triangle t(n,m) = A004001(m+1)+A004001(n-m+1)-A004001(n+1) read by rows, 0<=m<=n.

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A283525 Remainder when sum of first n terms of A004001 is divided by 3*n.

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A317921 a(1) = a(2) = 1; for n >= 3, a(n) = 3*a(t(n)) - a(n-t(n)) where t = A004001.

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A051284 a(n) is the number k, 2^n < k < 2^(n+1), such that k/c(k) is a minimum in the interval, where c(k) is Hofstadter-Conway sequence A004001.

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A087741 a(n) = 1+Abs[A000040[A004001[n]]-A004001[A000040[n]]].

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A089996 a(n) = primes generated by the function ( f[n_]=Floor[(A004001[n]*Prime[n])*Log[2]/(2*PrimePi[n]+1)]).

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A089996 a(n) = primes generated by the function ( f[n_]=Floor[(A004001[n]Prime[n])Log[2]/(2*PrimePi[n]+1)]).