A005132 -id:A005132 - cp's OEIS frontend

A108840 First step in Recamán's sequence A005132 at which there are n consecutive subtraction steps in a row.

4, 22, 109, 1360, 24404, 17695283, 6824557148

Offset: 1

Sergio Pimentel, Jul 25 2005

			E.g., 109 is the third term of this sequence because it the first step in Recamán's sequence at which there are three consecutive subtraction steps in a row:
109. 370 - 109 = 261
110. 261 - 110 = 151
111. 151 - 111 = 40

Eric Weisstein. Recamán's Sequence.

Cf. A005132, A064288.

a(6)-a(7) from Jud McCranie, Jan 24 2020

A169633 The odd terms in Recamán's sequence (A005132).

Original entry on oeis.org

1, 3, 7, 13, 21, 11, 23, 9, 25, 43, 63, 41, 17, 43, 15, 45, 79, 113, 77, 39, 79, 37, 81, 35, 83, 33, 85, 31, 87, 29, 89, 27, 91, 157, 225, 155, 227, 153, 75, 153, 73, 155, 71, 157, 69, 159, 67, 161, 65, 163, 265, 367, 263, 369, 261, 151, 265, 379, 495, 377, 259, 137, 261

Offset: 1

Eric Desbiaux, Mar 15 2010

nonn

More terms from R. J. Mathar, Oct 09 2010

A169749 A variation on Recamán's sequence A005132: see Comments for definition.

Original entry on oeis.org

1, 3, 6, 2, 7, 5, 11, 4, 12, 8, 17, 27, 16, 10, 22, 9, 23, 15, 30, 14, 31, 13, 32, 52, 42, 21, 43, 20, 44, 19, 45, 18, 46, 34, 63, 33, 64, 50, 82, 49, 83, 48, 84, 47, 85, 69, 51, 90, 70, 110, 88, 129, 87, 130, 86, 41, 87, 40, 88, 39, 89, 38, 90, 37, 91, 36, 92, 35, 93, 152, 128

Offset: 1

Rodolfo Kurchan, Apr 08 2010

nonn

Similar to A169748, but now B = 2,4,6,8,10,12,... (the even numbers).

Corrected and extended by D. S. McNeil, May 09 2010

A169755 A variation on Recamán's sequence A005132: see Comments for definition.

Original entry on oeis.org

1, 3, 2, 5, 9, 4, 10, 8, 15, 7, 16, 6, 17, 14, 26, 13, 27, 12, 28, 11, 29, 25, 20, 39, 19, 40, 18, 41, 35, 59, 34, 60, 33, 61, 32, 62, 31, 24, 56, 23, 57, 22, 58, 21, 29, 67, 106, 66, 107, 65, 108, 64, 55, 45, 90, 44, 91, 43, 92, 42, 93, 82, 30, 83, 71, 125, 70, 126, 69, 127, 68

Offset: 1

Rodolfo Kurchan, Apr 08 2010

nonn

We start with two sequences A and B. Here sequences A and B are both taken to be 1,2,3,4,5,6,7,....
We start with the first term of sequence A (which is 1).
To extend the sequence, we first try to subtract the next term of A from the current term, but that is allowed only if the result is a positive number not already in the sequence.
If that fails, we next try to subtract the next term of B from the current term, but again that is allowed only if the result is a positive number not already in the sequence.
If that fails, we next try to add the next term of A to the current term, but again that is allowed only if the result is a positive number not already in the sequence.
Finally, if that fails, we add the next term of B to the current term (this may produce repeated terms, but that is allowed at this step).

Cf. A005132, A169748-A169752.

Corrected and extended by D. S. McNeil, May 09 2010

A260251 Indices of terms in Recamán's sequence A005132 that do not appear for the first time.

Original entry on oeis.org

24, 26, 39, 41, 78, 80, 82, 84, 86, 112, 113, 125, 127, 130, 132, 137, 186, 188, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 224, 236, 238, 240, 242, 244, 246, 310, 312, 314, 316, 318, 320, 322, 324, 326, 328, 330, 332, 334, 336, 338, 353

Offset: 1

Gionata Neri, Apr 08 2016

nonn

A064284(A005132(a(n))) > 1.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A005132, A064284.

A269831 Least term of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 2, 6, 8, 14, 26, 4, 47, 92, 111, 181, 150, 371, 361, 781, 828, 366, 19

Offset: 1

Danny Rorabaugh, Mar 05 2016

The height (A064289) of a term in Recamán's sequence (A005132) = number of addition steps - number of subtraction steps to produce it.

Index entries for sequences related to Recamán's sequence

Cf. A064289, A064290, A064293, A269830, A269832.

A269832 Greatest term of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 3, 7, 13, 25, 46, 91, 164, 286, 515, 962, 1744, 3137, 5810, 10319, 18953, 35079, 63237

Offset: 1

Danny Rorabaugh, Mar 05 2016

The height (A064289) of a term in Recamán's sequence (A005132) = number of addition steps - number of subtraction steps to produce it.

Index entries for sequences related to Recamán's sequence

Cf. A064289, A064290, A064293, A269830, A269831.

A293273 a(n) is the smallest positive k <> n such that f(k) is divisible by f(n) where f = A005132, or 0 if no such k exists.

Original entry on oeis.org

2, 3, 8, 3, 9, 35, 43, 15, 20, 11, 28, 7, 32, 21, 83, 15, 69, 26, 152, 24, 116, 47, 44, 20, 48, 18, 43, 59, 30, 63, 20, 104, 41, 71, 39, 75, 72, 35, 35, 36, 33, 79, 92, 83, 96, 87, 100, 91, 245, 95, 239, 67, 276, 19, 119, 63, 109, 57, 103, 51, 185, 45, 139, 35, 145, 86, 415, 84, 192, 82, 184, 80, 180, 78, 176

Offset: 1

Altug Alkan, Oct 10 2017

Conjecture: a(n) > 0 for all n.

			a(6) = 35 because A005132(35) = 78 is divisible by A005132(6) = 13 and 78 is the smallest positive number which is not equal to 6 with this property.

Robert Israel, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of the first 100000 terms
Rémy Sigrist, Scatterplot of the first 100000 terms of the first difference

Cf. A005132, A057167.

N:= 10^4: # to use A005132(n) for n = 1..N
S:= {0}:
A5132:= Array(0..N):
A5132[0]:= 0:
for n from 1 to N do
  v:= A5132[n-1]-n;
  if v < 0 or member(v,S) then v:= A5132[n-1]+n fi;
  A5132[n]:= v;
  S:= S union {v};
od:
f:= proc(n) local k;
  for k from 1 to N do
    if k <> n and A5132[k] mod A5132[n] = 0 then return k fi
  od:
0
end proc:
Res:= NULL:
for n from 1 do
  v:= f(n);
  if v = 0 then break fi;
  Res:= Res,v;
od:
Res; # Robert Israel, Oct 10 2017

A329596 A variation of Recamán's sequence (A005132): a(0) = 0; a(1) = 1; a(2) = 3; for n > 2, a(n) = sopfr(a(n-1))- sopfr(n) if positive and not already in the sequence, otherwise a(n) = sopfr(a(n-1)) + sopfr(n).

Original entry on oeis.org

0, 1, 3, 6, 9, 11, 16, 15, 2, 8, 13, 24, 16, 21, 19, 27, 17, 34, 27, 28, 20, 19, 32, 33, 5, 15, 23, 14, 20, 38, 31, 62, 43, 29, 10, 19, 29, 66, 37, 53, 42, 53, 41, 84, 29, 18, 33, 61, 50, 26, 27, 29, 12, 60, 23, 7, 20, 31, 62, 92, 39, 77, 51, 33, 26, 33, 30, 77, 39, 42

Offset: 0

Bence Bernáth, Nov 17 2019

On the graph it seems that there are lines where the density of points is higher than elsewhere. These lines correspond to those which are easily observable on A001414. Up to n=100000 there are 13413 prime numbers in this sequence while at A001414 there are 21877; surely the primes are distributed differently in these sequences.

Regarding the MATLAB code: factor(0)=sum(factor(0))=0 and factor(1)=sum(factor(1))=1, this can be very misleading, attention needed during using sum(factor(n)) as sopfr(n).

			a(3)=6, factor(6)=[2 3], sum of factor(6) is 5. Then n=4, sum of factor(4) is 2+2=4. 5-4 = 1 but 1 is already in the sequence so a(4)=5+4=9.

Bence Bernáth, Table of n, a(n) for n = 0..10000
Bence Bernáth, Table of n, a(n) for n = 0..200000
Michael De Vlieger, First 65 terms drawn as a spiral, akin to the Harriss drawing at A005132.
Michael De Vlieger, video of first 128 steps of this sequence, with audio accompaniment generated by aspects of the sequence. Nov 19, 2019.

Cf. A001414, A005132.

length_seq=100000;
sequence(1)=0; %sum(factor(0))=0
sequence(2)=1; %sum(factor(1))=1
for i1=2:1:length_seq
if  (sum(factor(sequence(i1)))-sum(factor((i1))))>0 && (ismember((sum(factor(sequence(i1)))-sum(factor((i1)))),sequence)==0)
      sequence(i1+1)=(sum(factor(sequence(i1)))-sum(factor((i1))));
   else
       sequence(i1+1)=(sum(factor(sequence(i1)))+sum(factor((i1))));
end
end
result=transpose(sequence);

Block[{f}, f[n_] := Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[n]]; Nest[Append[#1, If[And[#3 >= 0, FreeQ[#1, #3]], #3, f@ #1[[-1]] + f@ #2]] & @@ {#1, #2, f@ #1[[-1]] - f@#2} & @@ {#, Length@ #} &, {0}, 69] ] (* Michael De Vlieger, Nov 19 2019 *)

A330918 Number of k such that A005132(k) > n for 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 4, 4, 4, 5, 5, 6, 7, 8, 8, 8, 8, 7, 7, 6, 7, 7, 8, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 16, 16, 16, 14, 13, 12, 12, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 29, 28, 26

Offset: 0

Seiichi Manyama, May 02 2020

			{A005132(n)} = {0, 1, 3, 6, ...}.
0 <= 2, 1 <= 2 and 3 > 2. So a(2) = 1.
0 <= 3, 1 <= 3, 3 <= 3 and 6 > 3 So a(3) = 1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Recamán's sequence

Cf. A005132.

cp's OEIS Frontend

A108840 First step in Recamán's sequence A005132 at which there are n consecutive subtraction steps in a row.

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A169633 The odd terms in Recamán's sequence (A005132).

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A169749 A variation on Recamán's sequence A005132: see Comments for definition.

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A169755 A variation on Recamán's sequence A005132: see Comments for definition.

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A260251 Indices of terms in Recamán's sequence A005132 that do not appear for the first time.

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A269831 Least term of height n in Recamán's sequence A005132.

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A269832 Greatest term of height n in Recamán's sequence A005132.

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A293273 a(n) is the smallest positive k <> n such that f(k) is divisible by f(n) where f = A005132, or 0 if no such k exists.

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Maple

A329596 A variation of Recamán's sequence (A005132): a(0) = 0; a(1) = 1; a(2) = 3; for n > 2, a(n) = sopfr(a(n-1))- sopfr(n) if positive and not already in the sequence, otherwise a(n) = sopfr(a(n-1)) + sopfr(n).

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MATLAB

Mathematica

A330918 Number of k such that A005132(k) > n for 0 <= k <= n.

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