A007379 -id:A007379 - cp's OEIS frontend

A002516 Earliest sequence with a(a(n)) = 2n.

0, 3, 6, 2, 12, 7, 4, 10, 24, 11, 14, 18, 8, 15, 20, 26, 48, 19, 22, 34, 28, 23, 36, 42, 16, 27, 30, 50, 40, 31, 52, 58, 96, 35, 38, 66, 44, 39, 68, 74, 56, 43, 46, 82, 72, 47, 84, 90, 32, 51, 54, 98, 60, 55, 100, 106, 80, 59, 62, 114, 104, 63, 116, 122, 192, 67, 70, 130

Offset: 0

Colin Mallows

T. D. Noe, Table of n, a(n) for n = 0..1000
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences of the a(a(n)) = 2n family

Cf. A002517, A004767, A007379, A017089, A091067.

import Data.List (transpose)
a002516 n = a002516_list !! n
a002516_list = 0 : concat (transpose
[a004767_list, f a002516_list, a017089_list, g $ drop 2 a002516_list])
where f [z] = []; f (_:z:zs) = 2 * z : f zs
g [z] = [z]; g (z:_:zs) = 2 * z : g zs
-- Reinhard Zumkeller, Jun 08 2015

a[0] = 0; a[n_ /; Mod[n, 2] == 0] := a[n] = 2*a[n/2]; a[n_ /; Mod[n, 4] == 1] := n+2; a[n_ /; Mod[n, 4] == 3] := 2(n-2); Table[a[n], {n, 0, 67}] (* Jean-François Alcover, Feb 06 2012, after Henry Bottomley *)

v2(n)=valuation(n,2)
a(n)=2^v2(n)*(-1+3/2*n/2^v2(n)-(-3+1/2*n/2^v2(n))*(-1)^((n/2^v2(n)-1)/2))

a(n)=local(t); if(n<1,0,if(n%2==0,2*a(n/2),t=(n-1)/2; 3*t+1/2-(t-5/2)*(-1)^t)) \\ Ralf Stephan, Feb 22 2004

a(4n) = 2*(a(2n)), a(4n+1) = 4n+3, a(4n+2) = 2*(a(2n+1)), a(4n+3) = 8n+2. - Henry Bottomley, Apr 27 2000
From Ralf Stephan, Feb 22 2004: (Start)
a(n) = n + 2*A006519(n) if odd part of n is of form 4k+1, or 2n - 4*A006519(n) otherwise.
a(2n) = 2*a(n), a(2n+1) = 2n + 3 + (2n - 5)*[n mod 2].
G.f.: Sum_{k>=0} 2^k*t(6t^6 + t^4 + 2t^2 + 3)/(1 - t^4)^2, t = x^2^k. (End)

A002517 Earliest sequence with a(a(n))=3n.

Original entry on oeis.org

0, 2, 3, 6, 5, 12, 9, 8, 21, 18, 11, 30, 15, 14, 39, 36, 17, 48, 27, 20, 57, 24, 23, 66, 63, 26, 75, 54, 29, 84, 33, 32, 93, 90, 35, 102, 45, 38, 111, 42, 41, 120, 117, 44, 129, 108, 47, 138, 51, 50, 147, 144, 53, 156, 81, 56, 165, 60, 59, 174, 171, 62, 183, 72, 65, 192

Offset: 0

Colin Mallows

a(3*n+1) = A016789(n), a(3*n+2) = A017197(n). - Reinhard Zumkeller, Jun 04 2015

Cf. A002516, A007379, A005843.

Cf. A007494 (sorted), A016789, A017197.

import Data.List (transpose)
a002517 n = a002517_list !! n
a002517_list = 0 : concat
   (transpose [[2, 5 ..], [3, 12 ..], map (* 3) $ tail a002517_list])
-- Reinhard Zumkeller, Jun 04 2015

a[n_] := a[n] = Which[ Mod[n, 3] == 0, 3*a[n/3], Mod[n, 3] == 1, n+1, True, 3*(n-1)]; Table[a[n], {n, 0, 65}] (* Jean-François Alcover, Sep 24 2012 *)

a(3n)=3*a(n), a(3n+1)=3n+2, a(3n+2)=9n+3

Formula and more terms from Henry Bottomley, Apr 27 2000

A002518 Earliest sequence with a(a(n))=5n.

Original entry on oeis.org

0, 2, 5, 4, 15, 10, 7, 30, 9, 40, 25, 12, 55, 14, 65, 20, 17, 80, 19, 90, 75, 22, 105, 24, 115, 50, 27, 130, 29, 140, 35, 32, 155, 34, 165, 150, 37, 180, 39, 190, 45, 42, 205, 44, 215, 200, 47, 230, 49, 240, 125, 52, 255, 54, 265, 60, 57, 280, 59, 290, 275, 62, 305

Offset: 0

Colin Mallows

Cf. A002516, A002517, A007379.

a[n_] := a[n] = Which[ Mod[n, 5] == 0, 5*a[n/5], Mod[n, 5] == 1, n+1, Mod[n, 5] == 2, 5*(n-2)+5, Mod[n, 5] == 3, n+1, True, 5*(n-4)+15]; a[0] = 0; Table[ a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 24 2012 *)

a(5n)=5*a(n), a(5n+1)=5n+2, a(5n+2)=25n+5, a(5n+3)=5n+4, a(5n+4)=25n+15

Corrected description and more terms from Henry Bottomley, Apr 27 2000

A054786 Earliest sequence with a(a(n)) = 6n.

Original entry on oeis.org

0, 2, 6, 4, 18, 7, 12, 30, 9, 48, 11, 60, 36, 14, 78, 16, 90, 19, 24, 102, 21, 120, 23, 132, 108, 26, 150, 28, 162, 31, 42, 174, 33, 192, 35, 204, 72, 38, 222, 40, 234, 43, 180, 246, 45, 264, 47, 276, 54, 50, 294, 52, 306, 55, 288, 318, 57, 336, 59, 348, 66, 62, 366, 64

Offset: 0

Henry Bottomley, Apr 27 2000

Cf. A002516, A002517, A002518, A007379.

a[0] = 0; a[n_] := a[n] = Switch[ Mod[n, 12], 0 | 6, 6*a[n/6], 1 | 3 | 8 | 10, n+1, 2 | 4 | 9 | 11, 6*n-6, 5, n+2, 7, 6*n-12]; Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Dec 20 2011, after formula *)

a(12n)=6*a(2n), a(12n+1)=12n+2, a(12n+2)=72n+6, a(12n+3)=12n+4, a(12n+4)=72n+18, a(12n+5)=12n+7, a(12n+6)=6*a(2n+1), a(12n+7)=72n+30, a(12n+8)=12n+9, a(12n+9)=72n+48, a(12n+10)=12n+11, a(12n+11)=72n+60.

Typo in formula corrected by Reinhard Zumkeller, Jul 23 2010

A054787 Earliest sequence with a(a(n))=7n.

Original entry on oeis.org

0, 2, 7, 4, 21, 6, 35, 14, 9, 56, 11, 70, 13, 84, 49, 16, 105, 18, 119, 20, 133, 28, 23, 154, 25, 168, 27, 182, 147, 30, 203, 32, 217, 34, 231, 42, 37, 252, 39, 266, 41, 280, 245, 44, 301, 46, 315, 48, 329, 98, 51, 350, 53, 364, 55, 378, 63, 58, 399, 60, 413, 62, 427, 392

Offset: 0

Henry Bottomley, Apr 27 2000

Cf. A002516, A002517, A002518, A007379.

a[n_] := a[n] = Which[ Mod[n, 7] == 0, 7*a[n/7], Mod[n, 7] == 1, n+1, Mod[n, 7] == 2, 7*(n-2)+7, Mod[n, 7] == 3, n+1, Mod[n, 7] == 4, 7*(n-4)+21, Mod[n, 7] == 5, n+1, Mod[n, 7] == 6, 7*(n-6)+35]; a[0] = 0; Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Sep 24 2012 *)

a(7n)=7*a(n), a(7n+1)=7n+2, a(7n+2)=49n+7, a(7n+3)=7n+4, a(7n+4)=49n+21, a(7n+5)=7n+6, a(7n+6)=49n+35

A054790 Earliest sequence with a(a(n))=10n.

Original entry on oeis.org

0, 2, 10, 4, 30, 6, 50, 8, 70, 11, 20, 90, 13, 120, 15, 140, 17, 160, 19, 180, 100, 22, 210, 24, 230, 26, 250, 28, 270, 31, 40, 290, 33, 320, 35, 340, 37, 360, 39, 380, 300, 42, 410, 44, 430, 46, 450, 48, 470, 51, 60, 490, 53, 520, 55, 540, 57, 560, 59, 580, 500, 62, 610

Offset: 0

Henry Bottomley, Apr 27 2000

Index entries for sequences of the a(a(n)) = 2n family

Cf. A002516, A002517, A002518, A007379.

a[0] = 0; a[n_] := Which[m = Mod[n, 20]; m == 0, 10*n-100, m == 9, n+2, m == 10, n+10, m == 11, 10*n-20, MemberQ[ {2, 4, 6, 8, 13, 15, 17, 19}, m], 10*n-10, True, n+1]; Table[ a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 24 2012 *)

A054793 Earliest sequence with a(a(n)) = n^4.

Original entry on oeis.org

0, 1, 3, 16, 5, 256, 7, 1296, 9, 4096, 11, 10000, 13, 20736, 15, 38416, 81, 18, 83521, 20, 130321, 22, 194481, 24, 279841, 26, 390625, 28, 531441, 30, 707281, 32, 923521, 34, 1185921, 36, 1500625, 38, 1874161, 40, 2313441, 42, 2825761, 44, 3418801, 46

Offset: 0

Henry Bottomley, Apr 27 2000

Cf. A000290, A007379.

a[n_] := a[n] = Which[r = n^(1/4); IntegerQ[r], a[r]^4, OddQ[n - Floor[r]^4], n+1, True, (n-1)^4]; a[0]=0; a[1]=1; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Aug 07 2012, after formula *)

from sympy import integer_nthroot
def A054793(n):
    a, b = integer_nthroot(n,4)
    return n if n <= 1 else A054793(a)**4 if b else n+1 if (n-a**4) % 2 else (n-1)**4 # Chai Wah Wu, Apr 02 2021

if n is a 4th power then a(n)=a(n^(1/4))^4, otherwise if the difference between n and the highest 4th power less than n is odd then a(n)=n+1, otherwise a(n)=(n-1)^4.

A054788 Earliest sequence with a(a(n))=8n.

Original entry on oeis.org

0, 2, 8, 4, 24, 6, 40, 9, 16, 56, 11, 80, 13, 96, 15, 112, 64, 18, 136, 20, 152, 22, 168, 25, 32, 184, 27, 208, 29, 224, 31, 240, 192, 34, 264, 36, 280, 38, 296, 41, 48, 312, 43, 336, 45, 352, 47, 368, 320, 50, 392, 52, 408, 54, 424, 57, 72, 440, 59, 464, 61, 480, 63, 496

Offset: 0

Henry Bottomley, Apr 27 2000

Index entries for sequences of the a(a(n)) = 2n family

Cf. A002516, A002517, A002518, A007379.

nmax = 63; amax = 8*nmax; t = {{0, a[0] = 0}, {1, a[1] = 2}, {2, a[2]}}; While[ !FreeQ[t, a], t = Table[{n, a[n]}, {n, 0, nmax}]; n = Select[t, !IntegerQ[ #[[2]] ] &, 1][[1, 1]]; t2 = Union[ Flatten[ Append[ Select[ t, IntegerQ[ #[[2]] ] &], n]]]; an = If[n == 2, 8, Select[ Complement[ Range[ Max[t2] ], t2], Mod[#, 8] != 0 &, 1][[1]] ]; a[n] = an; While[ an < amax, an = a[n = an] = 8 n]]; Table[ a[n], {n, 0, nmax}] (* Jean-François Alcover, Jan 11 2012 *)

A054789 Earliest sequence with a(a(n)) = 9n.

Original entry on oeis.org

0, 2, 9, 4, 27, 6, 45, 8, 63, 18, 11, 90, 13, 108, 15, 126, 17, 144, 81, 20, 171, 22, 189, 24, 207, 26, 225, 36, 29, 252, 31, 270, 33, 288, 35, 306, 243, 38, 333, 40, 351, 42, 369, 44, 387, 54, 47, 414, 49, 432, 51, 450, 53, 468, 405, 56, 495, 58, 513, 60, 531, 62, 549

Offset: 0

Henry Bottomley, Apr 27 2000

Index entries for sequences of the a(a(n)) = 2n family

Cf. A002516, A002517, A002518, A007379, A008585.

a[0] = 0; a[n_] := Which[m = Mod[n, 18]; m == 0, 9*n-81, m == 9, n+9, MemberQ[ {1, 3, 5, 7, 10, 12, 14, 16}, m], n+1, True, 9*n-9]; Table[ a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 24 2012 *)

cp's OEIS Frontend

A002516 Earliest sequence with a(a(n)) = 2n.

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A002517 Earliest sequence with a(a(n))=3n.

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A002518 Earliest sequence with a(a(n))=5n.

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A054786 Earliest sequence with a(a(n)) = 6n.

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A054787 Earliest sequence with a(a(n))=7n.

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A054790 Earliest sequence with a(a(n))=10n.

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A054793 Earliest sequence with a(a(n)) = n^4.

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A054788 Earliest sequence with a(a(n))=8n.

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