A099506 -id:A099506 - cp's OEIS frontend

A099507 a(n) is such that A099506(a(n)) = n.

1, 4, 2, 6, 17, 3, 10, 5, 12, 7, 14, 53, 9, 8, 11, 22, 13, 24, 15, 26, 99, 103, 28, 107, 30, 19, 32, 111, 16, 23, 18, 25, 42, 20, 29, 46, 31, 48, 33, 50, 209, 35, 211, 56, 37, 58, 217, 39, 60, 21, 41, 64, 43, 66, 245, 45, 72, 47, 74, 49, 27, 51, 34, 267, 80, 36, 82, 277, 38, 57

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 20 2004

a(199) > 10^6 if it exists. - Robert Israel, Jun 17 2015

			Gives the positions in the sequence A099506 at which the integers n>0 appear. That sequence begins: 1,3,6,2,8,4,... so a(1)=1, a(2)=4, a(3)=2, etc.

Robert Israel, Table of n, a(n) for n = 1..198
Robert Israel, Table of n, a(n) for n = 1..10000 with 0 for entries greater than 10^6 (if they exist at all)

Cf. A099506.

N = 10^6;
M = 10*N;  % search up to A099506(N) while A099507 < M
% returns 0 for entries not found
B = zeros(1, M);
A = zeros(1, N);
A(1) = 1;
B(1) = 1;
for n = 2:N
  t = mod(-A(n-1)-1,n)+1;
  bm = t + [0:floor((M-t)/n)]*n;
  bz = find(B(bm)==0,1);
  if numel(bz)==0
      break
  end
  m = t + n*(bz-1);
  A(n) = m;
  B(m) = n;
end;
B(1:1000) % Robert Israel, Jun 17 2015

A099518 RECORDS transform of A099506.

Original entry on oeis.org

1, 3, 6, 8, 10, 14, 15, 17, 19, 29, 31, 34, 50, 61, 63, 66, 69, 72, 79, 94, 96, 124, 131, 192, 225, 300, 353, 399, 433, 623, 632, 636, 795, 799, 806, 813, 1027, 1092, 1419, 1436, 1440, 1448, 1483, 1875, 2236, 2244, 2450, 2494, 2776, 2827, 3253, 3494, 4113, 4491

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 20 2004

			A099506 starts with: 1,3,6,2,8,4,10,14,13,7,15,... so the records are 1,3,6,8,10,14,15,...

Cf. A099506, A099507.

A099519 Positions in A099506 of the values in the RECORDS transform of that sequence (A099518).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 15, 16, 18, 20, 21, 27, 34, 36, 38, 40, 44, 52, 54, 55, 59, 68, 101, 108, 128, 181, 197, 224, 281, 283, 288, 359, 363, 367, 368, 493, 510, 643, 645, 649, 665, 676, 800, 804, 876, 892, 990, 1008, 1017, 1250, 1476, 1614, 1676, 2140, 2286

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 20 2004

			Sequence A099506 is 1,3,6,2,8,4,10,14,13,7,15,... so the records are (A099518) 1,3,6,8,10,14,15,... which appear in positions 1,2,3,5,7,8,11,....

Cf. A099506, A099507, A099518.

A099506(a(n))=A099518(n).

A125717 a(0)=0; thereafter a(n) = the smallest nonnegative integer not already in the sequence such that a(n-1) is congruent to a(n) (mod n).

Original entry on oeis.org

0, 1, 3, 6, 2, 7, 13, 20, 4, 22, 12, 23, 11, 24, 10, 25, 9, 26, 8, 27, 47, 5, 49, 72, 48, 73, 21, 75, 19, 77, 17, 79, 15, 81, 115, 45, 117, 43, 119, 41, 121, 39, 123, 37, 125, 35, 127, 33, 129, 31, 131, 29, 133, 80, 134, 189, 245, 74, 16, 193, 253, 70, 132, 69, 197, 67, 199, 65

Offset: 0

Leroy Quet, Feb 01 2007

This sequence seems likely to be a permutation of the nonnegative integers.
A245340(n) = smallest m such that a(m) = n, or -1 if n never appears.
See A245394 and A245395 for record values of a(n) and where they occur. - Reinhard Zumkeller, Jul 21 2014
See A370956 and A370959 for record values of the inverse A245340 and where they occur. - N. J. A. Sloane, Apr 29 2024
A very nice (maybe the most natural) variant of Recamán's sequence A005132. - M. F. Hasler, Nov 03 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..100000 (first 10000 terms from Ferenc Adorján)
Ferenc Adorján, Some characteristics of Leroy Quet's permutation sequences
N. J. A. Sloane, Log-log plot of A370956 vs A370959 (shows terms in A125717 that take the longest to appear).
Index entries for sequences related to Recamán's sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A245340 (inverse), A370957 (first differences), A245394 & A245395 (records in this sequence), A370956 & A370959 (records in inverse).

See also A005132 (Recaman), A099506, A125715, A125718, A125725.

import Data.IntMap (singleton, member, (!), insert)
a125717 n = a125717_list !! n
a125717_list =  0 : f [1..] 0 (singleton 0 0) where
   f (v:vs) w m = g (reverse[w-v,w-2*v..1] ++ [w+v,w+2*v..]) where
     g (x:xs) = if x `member` m then g xs else x : f vs x (insert x v m)
-- Reinhard Zumkeller, Jul 21 2014

f[l_List] := Block[{n = Length[l], k = Mod[l[[ -1]], n]},While[MemberQ[l, k], k += n];Append[l, k]];Nest[f, {0}, 70] (* Ray Chandler, Feb 04 2007, updated for change to offset Oct 10 2019 *)

{Quet_p2(n)=/* Permutation sequence a'la Leroy Quet, A125717 */local(x=[1],k=0,w=1); for(i=2,n,if((k=x[i-1]%i)==0,k=i);while(bittest(w,k-1)>0,k+=i);x=concat(x,k);w+=2^(k-1)); return(x)} \\ Ferenc Adorjan

A125717(n,show=0)={my(u=1,a);for(n=1,n,a%=n;while(bittest(u,a),a+=n);u+=1<M. F. Hasler, Nov 03 2014

from itertools import count, islice
def agen(): # generator of terms
    an, aset = 0, {0}
    for n in count(1):
        yield an
        an = next(m for m in count(an%n, n) if m not in aset)
        aset.add(an)
print(list(islice(agen(), 70))) # Michael S. Branicky, Jun 07 2023

Extended by Ray Chandler, Feb 04 2007
a(0) added by Franklin T. Adams-Watters, Mar 31 2014
Edited by N. J. A. Sloane, Mar 15 2024

A099520 a(1)=1; for n>1, a(n)=smallest integer m>0 not yet appearing in sequence such that m+a(n-1) is a multiple of n-1.

Original entry on oeis.org

1, 2, 4, 5, 3, 7, 11, 10, 6, 12, 8, 14, 22, 17, 25, 20, 28, 23, 13, 44, 16, 26, 18, 51, 21, 29, 49, 32, 24, 34, 56, 37, 27, 39, 63, 42, 30, 81, 33, 45, 35, 47, 79, 50, 38, 52, 40, 54, 90, 57, 43, 59, 97, 9, 99, 66, 46, 68, 48, 70, 110, 73, 113, 76, 116, 144, 120, 148, 124, 83

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 20 2004

			a(1)=1. a(2)=2 because a(2)+a(1)=2+1=3 which is a multiple of 1. Could not use a(2)=1 since 1 has already been used. a(3)=4 so that a(3)+a(2)=4+2=6 which is a multiple of 2, etc.

Cf. A099506 for similar sequence with a(n)+a(n-1)=multiple of n.

A371214 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(n-1) + a(n) is a multiple of n, and the least value not yet in the sequence appears as soon as possible.

Original entry on oeis.org

0, 1, 7, 2, 22, 3, 45, 4, 76, 5, 115, 6, 18, 8, 216, 9, 279, 10, 350, 11, 429, 12, 516, 13, 611, 14, 714, 15, 825, 16, 944, 17, 47, 19, 1205, 20, 1348, 21, 55, 23, 1657, 24, 1824, 25, 1999, 26, 2182, 27, 2373, 28, 2572, 29, 2779, 30, 2994, 31, 3217, 32, 3448

Offset: 0

Rémy Sigrist, Mar 15 2024

nonn

To build the sequence:
- we start with a(0) = 0, and repeatedly:
- let a(n) be the last known term and v the least value not yet in the sequence,
- if a(n) + v is a multiple of n+1 then a(n+1) = v,
- otherwise a(n+2) = v and a(n+1) is chosen as small as possible in such a way as to satisfy the required congruences (this is always possible as n+1 and n+2 are coprime).
The construction is similar to that of A367288.
This sequence is a variant of A099506 and, by design, is guaranteed to be a permutation of the nonnegative integers (with inverse A371215).

			The first terms are:
  n   a(n)  (a(n-1) + a(n))/n
  --  ----  -----------------
   0     0  N/A
   1     1                  1
   2     7                  4
   3     2                  3
   4    22                  6
   5     3                  5
   6    45                  8
   7     4                  7
   8    76                 10
   9     5                  9
  10   115                 12
  11     6                 11
  12    18                  2

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A099506, A367288, A371215 (inverse).

PARI
```
See Links section.
```

A294734 Lexicographically earliest sequence of distinct positive odd numbers such that, for any n > 1, n divides a(n-1) + a(n).

Original entry on oeis.org

1, 3, 9, 7, 13, 5, 23, 17, 19, 11, 33, 15, 37, 47, 43, 21, 81, 27, 49, 31, 53, 35, 57, 39, 61, 69, 93, 75, 41, 79, 45, 51, 147, 91, 119, 25, 123, 29, 127, 73, 173, 121, 137, 83, 97, 87, 101, 139, 155, 95, 109, 99, 113, 103, 117, 107, 235, 55, 63, 177, 67, 181

Offset: 1

Rémy Sigrist, Nov 07 2017

nonn

This sequence is a variant of A099506 and of A134204.
The variant where "n divides a(n) + a(n+1)" corresponds to A005408 (the odd numbers).
Empirically, the scatterplot of the first terms shows a bundle of curves that correspond to terms sharing the same value of Sum_{k=2.. n} ((-1)^k (a(k+1) + a(k))/k) (see Links section); the sequence A134204 has similar features.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 10000 terms
Rémy Sigrist, Colored scatterplot of the first 100000 terms
Rémy Sigrist, PARI program for A294734

Cf. A005408, A099506, A134204.

PARI
```
See Links section.
```

A099521 a(n) is such that A099520(a(n))=n.

Original entry on oeis.org

1, 2, 5, 3, 4, 9, 6, 11, 54, 8, 7, 10, 19, 12, 72, 21, 14, 23, 84, 16, 25, 13, 18, 29, 15, 22, 33, 17, 26, 37, 156, 28, 39, 30, 41, 190, 32, 45, 34, 47, 218, 36, 51, 20, 40, 57, 42, 59, 27, 44, 24, 46, 284, 48, 286, 31, 50, 292, 52, 77, 300, 79, 35, 81, 85, 56, 87, 58, 89, 60

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 20 2004

			Gives the positions in the sequence A099520 at which the integers n>0 appear. A099520 begins: 1,2,4,5,3,7,11,10,6,12,8,..., so the positions of the integers are: 1,2,5,3,4,9,6,...

Cf. A099520. Also A099506 and A099507 for sequences based on a slightly different definition.

cp's OEIS Frontend

A099507 a(n) is such that A099506(a(n)) = n.

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A099518 RECORDS transform of A099506.

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A099519 Positions in A099506 of the values in the RECORDS transform of that sequence (A099518).

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Formula

A125717 a(0)=0; thereafter a(n) = the smallest nonnegative integer not already in the sequence such that a(n-1) is congruent to a(n) (mod n).

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A099520 a(1)=1; for n>1, a(n)=smallest integer m>0 not yet appearing in sequence such that m+a(n-1) is a multiple of n-1.

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A371214 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(n-1) + a(n) is a multiple of n, and the least value not yet in the sequence appears as soon as possible.

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A294734 Lexicographically earliest sequence of distinct positive odd numbers such that, for any n > 1, n divides a(n-1) + a(n).

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A099521 a(n) is such that A099520(a(n))=n.

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