A004001 -id:A004001 - cp's OEIS frontend

A087740 a(n) = 1 + abs(A004001(A005185(n)) - A005185(A004001(n))).

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

Offset: 1

Roger L. Bagula, Oct 01 2003

A "commutator" between the Hofstadter A005185 sequence and the Conway-Hofstadter A004001 sequence.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A004001, A005185, A284019 (compare the scatter plots).

Cf. also A286560.

Conway[n_Integer?Positive] := Conway[n] =Conway[Conway[n-1]] + Conway[n - Conway[n-1]] Conway[1] = Conway[2] = 1 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[Conway[Hofstadter[n]]-Hofstadter[Conway[n]]], {n, 1, digits}]

(define (A087740 n) (+ 1 (abs (- (A004001 (A005185 n)) (A005185 (A004001 n)))))) ;; Scheme-code for A004001 and A005185 given under those entries.

Data section extended to 120 terms by Antti Karttunen, May 22 2017

A088493 a(n) = Sum_{k=1..8} floor(p(n, k)/p(n-1, k)), where p(n, k) = n!/( Product_{i=1..floor(n/2^k)} A004001(i) ).

Original entry on oeis.org

16, 24, 32, 40, 45, 56, 60, 72, 73, 88, 81, 104, 101, 120, 108, 136, 129, 152, 129, 168, 157, 184, 141, 200, 185, 216, 178, 232, 213, 248, 188, 264, 241, 280, 226, 296, 269, 312, 222, 328, 297, 344, 273, 360, 325, 376, 237, 392, 353, 408, 321, 424, 381, 440

Offset: 2

Roger L. Bagula, Nov 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A004001, A005185, A005229, A088491.

Conway[n_]:= Conway[n]= If[n<3, 1, Conway[Conway[n-1]] +Conway[n-Conway[n-1]]];
f[n_, k_]:= f[n, k]= Product[Conway[i], {i, Floor[n/2^k]}];
a[n_]:= a[n]= Sum[Floor[n*f[n-1,k]/f[n,k]], {k,8}];
Table[a[n], {n, 2, 70}] (* modified by G. C. Greubel, Mar 27 2022 *)

@CachedFunction
def b(n): # A004001
    if (n<3): return 1
    else: return b(b(n-1)) + b(n-b(n-1))
def f(n,k): return product( b(j) for j in (1..(n//2^k)) )
def A088493(n): return sum( (n*f(n-1,k)//f(n,k)) for k in (1..8) )
[A088493(n) for n in (2..70)] # G. C. Greubel, Mar 27 2022

a(n) = Sum_{k=1..8} floor(p(n, k)/p(n-1, k)), where p(n, k) = n!/( Product_{i=1..floor(n/2^k)} A004001(i) ).

Edited by G. C. Greubel, Mar 27 2022

A095902 Number of odd entries in A004001 that are <= 2^n.

Original entry on oeis.org

1, 2, 2, 3, 6, 12, 27, 55, 115, 235, 490, 994, 2008, 4036, 8120, 16280, 32640, 65344, 130879, 261935, 524057, 1048301, 2096855, 4193951, 8388239, 16776799, 33554339, 67109539, 134220995

Offset: 0

Robert G. Wilson v, Jun 12 2004

Even entries and odd entries are equal only when n=4, 6 and 12. Past that, the evens outnumber the odds.

Cf. A000079, A004001, A095901, A283480.

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; c = 0; k = 1; Do[ While[k <= 2^n, If[ Mod[ a[k], 2] == 1, c++ ]; k++ ]; Print[c], {n, 21}]

(define (A095902 n) (A283480 (A000079 n))) ;; Antti Karttunen, Mar 21 2017

a(n) = A283480(2^n) - Antti Karttunen, Mar 21 2017

a(22)-a(28) from Donovan Johnson, Jan 28 2009

A120474 Second differences of A004001.

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, Jul 06 2006

sign

Takes values in {0,1,-1}.

R. J. Mathar, Table of n, a(n) for n = 1..9998

Cf. A004001. First differences of A093879.

Differences[#, 2] &@ Nest[Append[#1, #1[[#1[[-1]] ]] + #1[[#2 - #1[[-1]] ]] ] & @@ {#, Length@ # + 1} &, {1, 1}, 105] (* Michael De Vlieger, Nov 05 2018 *)

a(n) = A004001(n+2) - 2*A004001(n+1) + A004001(n) = A093879(n+1) - A093879(n).

Edited by N. J. A. Sloane, Oct 01 2006

A172452 Partial products of A004001.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 48, 192, 768, 3840, 23040, 161280, 1128960, 9031680, 72253440, 578027520, 4624220160, 41617981440, 416179814400, 4577977958400, 54935735500800, 659228826009600, 8569974738124800, 119979646333747200, 1679715048672460800, 25195725730086912000, 377935885951303680000

Offset: 0

Roger L. Bagula, Feb 03 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

f[n_]:= f[n]= If[n<3, Fibonacci[n], f[f[n-1]] + f[n-f[n-1]]]; (* f = A004001 *)
a[n_]:= Product[f[j], {j,n}];
Table[a[n], {n,0,35}] (* modified by G. C. Greubel, Apr 27 2021 *)

@CachedFunction
def b(n): return fibonacci(n) if (n<3) else b(b(n-1)) + b(n-b(n-1)) # b=A004001
def a(n): return product(b(j) for j in (1..n))
[a(n) for n in (0..35)] # G. C. Greubel, Apr 27 2021

Definition simplified - The Assoc. Editors of the OEIS, Feb 24 2010

More terms added by G. C. Greubel, Apr 27 2021

A266348 a(1) = 1; for n > 1, a(n) = A004001(n+1) - A072376(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 3, 4, 4, 4, 4, 1, 2, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 1, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, 11, 11, 12, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 15, 16, 16, 16, 17, 18, 19, 19, 20, 21, 21

Offset: 1

Antti Karttunen, Jan 22 2016

When the terms are arranged as successively larger batches of 2^n, the terms A(n,k), k = 1 .. 2^n, on row n give the cumulative number of 1's encountered since the beginning of the row n of similarly organized irregular table A265754, up to and including the k-th term on that row:
1;
1, 1;
1, 2, 2, 2;
1, 2, 3, 3, 4, 4, 4, 4;
1, 2, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8;
...

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A000523, A004001, A072376, A265754.

lim = 100; b[1] = 1; b[2] = 1; b[n_] := b[n] = b[b[n - 1]] + b[n - b[n - 1]]; s = CoefficientList[Series[1/(2 - 2 x) (2 x - x^2 + Sum[ 2^(k - 1) x^2^k, {k, Floor@ Log2@ lim}]), {x, 0, lim}], x]; {1}~Join~Table[b[n + 1] - s[[n + 1]], {n, 2, lim}] (* Michael De Vlieger, Jan 26 2016, after Robert G. Wilson v at A004001 *)

(define (A266348 n) (if (= 1 n) 1 (- (A004001 (+ 1 n)) (A072376 n))))

a(1) = 1; for n > 1, a(n) = A004001(n+1) - A072376(n) = A004001(n+1) - 2^(A000523(n)-1).

A266399 a(n) = A188163(A088359(n)); positions where A004001 obtains unique values.

Original entry on oeis.org

5, 9, 10, 17, 18, 19, 22, 33, 34, 35, 36, 39, 40, 43, 49, 65, 66, 67, 68, 69, 72, 73, 74, 77, 78, 81, 87, 88, 91, 97, 107, 129, 130, 131, 132, 133, 134, 137, 138, 139, 140, 143, 144, 145, 148, 149, 152, 158, 159, 160, 163, 164, 167, 173, 174, 177, 183, 193, 194, 197, 203, 213, 228, 257, 258, 259, 260, 261

Offset: 1

Antti Karttunen, Jan 18 2016

nonn

Numbers n for which A004001(n-1) < A004001(n) < A004001(n+1).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A004001.
Subsequence of A088359 and A188163.
Cf. also A266188.

a(n) = A188163(A088359(n)) = A088359(A088359(n)-1) = A188163(A188163(1+n)).
Other identities. For all n >= 1:
A004001(a(n)) = A088359(n).

A266640 Reversed reduced frequency counts for A004001: a(n) = A265754(A054429(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 22 2016

Deleting all 1's and decrementing the remaining terms by one gives the sequence back.

			Illustration how the sequence can be constructed by concatenating the reversed reduced frequency counts R_n of each successive level n of A004001-tree:
                              1
                             / \
                            2   1
                           /|\   \
              ____________3 2 1   1
             /    /    /  | |\ \   \
    ________4  __3    2   1 2 1 1   1
   / / / / /  / /|   /|   | |\ \ \   \
  5 4 3 2 1  3 2 1  2 1   1 2 1 1 1   1
etc.

Antti Karttunen, Table of n, a(n) for n = 1..8191
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

Cf. A004001, A054429.
Cf. A000079 (positions of records, where n appears for the first time).
Cf. A265754 (obtained from the mirror image of the same tree).

(define (A266640 n) (A265754 (A054429 n)))

a(n) = A265754(A054429(n)).
Other identities. For all n >= 0:
a(2^n) = n+1.

A267103 Row 3 of A265903; numbers that occur exactly three times in A004001.

Original entry on oeis.org

4, 15, 27, 30, 48, 54, 57, 61, 86, 96, 102, 105, 112, 115, 119, 124, 157, 172, 182, 188, 191, 202, 208, 211, 218, 221, 225, 233, 236, 240, 245, 251, 293, 314, 329, 339, 345, 348, 364, 374, 380, 383, 394, 400, 403, 410, 413, 417, 429, 435, 438, 445, 448, 452, 460, 463, 467, 472, 481, 484, 488, 493, 499, 506, 558

Offset: 1

Antti Karttunen, Jan 18 2016

nonn

Numbers n for which A051135(n) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Column 3 of A265901, row 3 of A265903.

Cf. A004001, A051135, A087686, A188163, A266109.

a(n) = A087686(1+A266109(n)) = A087686(1+A087686(1+A188163(n))).

A269851 a(0) = 1, a(A087686(1+n)) = 2*a(n), a(A088359(n)) = A250469(a(n)), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001.

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 5, 6, 16, 21, 25, 7, 18, 15, 10, 12, 32, 45, 55, 49, 11, 42, 51, 35, 50, 27, 14, 36, 33, 30, 20, 24, 64, 93, 115, 91, 121, 13, 90, 123, 125, 77, 110, 147, 65, 98, 39, 22, 84, 105, 85, 102, 87, 70, 100, 57, 54, 28, 72, 69, 66, 60, 40, 48, 128, 189, 235, 203, 187, 169, 17, 186, 267, 305, 217, 143, 230

Offset: 0

Antti Karttunen, Mar 07 2016

nonn

Permutation of natural numbers obtained from the sieve of Eratosthenes, combined with the permutation obtained from Hofstadter-Conway $10000 sequence (A004001). Note the indexing: Domain starts from 0, range from 1.

Inverse: A269852.
Cf. A004001, A087686, A088359, A093879, A250469.
Related or similar permutations: A252755, A267111, A269855.

a(0) = 1, a(1) = 2, for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 2*a(n - A004001(n)), otherwise [when n is in A088359], a(n) = A250469(a(A004001(n)-1)).
As a composition of related permutations:
a(n) = A252755(A267111(n)).
Other identities. For all n >= 0:
a(2^n) = 2^(n+1).

cp's OEIS Frontend

A087740 a(n) = 1 + abs(A004001(A005185(n)) - A005185(A004001(n))).

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A088493 a(n) = Sum_{k=1..8} floor(p(n, k)/p(n-1, k)), where p(n, k) = n!/( Product_{i=1..floor(n/2^k)} A004001(i) ).

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A095902 Number of odd entries in A004001 that are <= 2^n.

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A120474 Second differences of A004001.

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A172452 Partial products of A004001.

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A266348 a(1) = 1; for n > 1, a(n) = A004001(n+1) - A072376(n).

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A266399 a(n) = A188163(A088359(n)); positions where A004001 obtains unique values.

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A266640 Reversed reduced frequency counts for A004001: a(n) = A265754(A054429(n)).

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