A125717 -id:A125717 - cp's OEIS frontend

A245340 Smallest m such that A125717(m) = n, or -1 if n never appears.

0, 1, 4, 2, 8, 21, 3, 5, 18, 16, 14, 12, 10, 6, 1518, 32, 58, 30, 184, 28, 7, 26, 9, 11, 13, 15, 17, 19, 102, 51, 100, 49, 98, 47, 96, 45, 94, 43, 92, 41, 90, 39, 88, 37, 86, 35, 84, 20, 24, 22, 505, 81, 2510, 79, 166, 77, 296, 75, 501, 73, 162, 71, 498, 69

Offset: 0

Reinhard Zumkeller, Jul 21 2014

nonn

Conjecture: a(n) is never -1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Ferenc Adorjan, 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).

Cf. A245394, A245395, A057167.

For RECORDS see A370956 and A370959.

import Data.IntMap (singleton, member, (!), insert)
a245340 n = a245340_list !! n
a245340_list = 0 : f [1..] [1..] 0 (singleton 0 0) where
   f us'@(u:us) vs'@(v:vs) w m
     | u `member` m = (m ! u) : f us vs' w m
     | otherwise    = 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 f us' vs x $ insert x v m

from itertools import count
def A245340(n):
    a, aset = 0, set()
    for m in count(1):
        if a==n: return m-1
        aset.add(a)
        a = next(a for a in count(a%m,m) if a not in aset) # Chai Wah Wu, Mar 13 2024

A245394 Record values in A125717.

Original entry on oeis.org

0, 1, 3, 6, 7, 13, 20, 22, 23, 24, 25, 26, 27, 47, 49, 72, 73, 75, 77, 79, 81, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 134, 189, 245, 253, 298, 341, 448, 454, 460, 525, 540, 546, 552, 558, 561, 734, 759, 769, 883, 892, 893, 903, 909, 915, 921, 927

Offset: 1

Reinhard Zumkeller, Jul 21 2014

nonn

a(n) = A125717(A245395(n)) and A125717(m) < a(n) for m < A245395(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..2562, all records < 10^8

Cf. A125717, A245395, A245340, A064227.

a245394 n = a245394_list !! (n-1)
(a245394_list, a245395_list) =  unzip $ f [0..] a125717_list (-1) where
   f (x:xs) (y:ys) r = if y > r then (y,x) : f xs ys y else f xs ys r

A245395 Where record values occur in A125717.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 19, 20, 22, 23, 25, 27, 29, 31, 33, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 60, 83, 106, 107, 111, 115, 148, 157, 161, 165, 169, 172, 173, 186, 192, 250, 256, 258, 264, 268, 272, 276, 280, 284, 287, 289

Offset: 1

Reinhard Zumkeller, Jul 21 2014

nonn

A245394(n) = A125717(a(n)) and A125717(m) < A245394(n) for m < a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..2562, positions of record values < 10^8

Cf. A125717, A245394, A245340, A064227.

a245395 n = a245395_list !! (n-1) -- a245395_list is defined in A245394.

A370957 First differences of A125717.

Original entry on oeis.org

1, 2, 3, -4, 5, 6, 7, -16, 18, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, 20, -42, 44, 23, -24, 25, -52, 54, -56, 58, -60, 62, -64, 66, 34, -70, 72, -74, 76, -78, 80, -82, 84, -86, 88, -90, 92, -94, 96, -98, 100, -102, 104, -53, 54, 55, 56, -171, -58, 177, 60, -183, 62, -63, 128, -130, 132, -134, 136, -138, 140, -142

Offset: 1

N. J. A. Sloane, Mar 15 2024

sign

In other words, a(n) = A125717(n)-A125717(n-1).

N. J. A. Sloane, Table of n, a(n) for n = 1..100000

Cf. A125717, A370958.

A370958 Normalized first differences of A125717.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, -2, 2, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -2, 2, 1, -1, 1, -2, 2, -2, 2, -2, 2, -2, 2, 1, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -1, 1, 1, 1, -3, -1, 3, 1, -3, 1, -1, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, 1, -3, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2

Offset: 1

N. J. A. Sloane, Mar 15 2024

sign

N. J. A. Sloane, Table of n, a(n) for n = 1..100000

Cf. A125717, A370957.

a(n) = A370957(n)/n.

A372057 After A125717(n) has been found, a(n) is the smallest missing number.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14

Offset: 0

N. J. A. Sloane, Apr 30 2024

nonn

Cf. A125717.

A370959 lists the distinct terms in order, and the first differences of A370956 specify the lengths of the runs of identical terms.

A364054 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

Original entry on oeis.org

1, 3, 6, 11, 4, 15, 2, 19, 38, 61, 32, 63, 26, 67, 24, 71, 18, 77, 16, 83, 12, 85, 164, 81, 170, 73, 174, 277, 384, 57, 283, 29, 160, 23, 162, 13, 315, 158, 321, 154, 327, 148, 329, 138, 331, 134, 333, 122, 345, 118, 347, 114, 353, 112, 363, 106, 369, 100, 371, 94, 375, 92, 385

Offset: 1

Ali Sada, Oct 19 2023

5 is the smallest positive integer missing from the first 1000 terms. Also in the interval a(100) to a(1000) there are no entries less than 100. (From W. Edwin Clark via SeqFan.)
Comments from N. J. A. Sloane, Oct 22 2023 (Start)
It appears that the graph of this sequence is dominated by pairs of diverging lines, as suggested by the sketch (see link). For example, around step n = 4619, a descending line is changing to a descending line around a(4619) = 65, a companion ascending line is coming to an end near a(4594) = 44518, and a strong ascending line is starting up around a(4620) = 88899.
It would be nice to have more terms, in order to get better estimates of the times t_i where these transitions happen, and heights alpha_i, beta_i, gamma_i where line breaks are.
The only well-defined points are the (t_i, alpha_i) where the descending lines end, as can be seen from the b-file, where the end point a(4619) = 65 is well-defined. The other transitions, where an ascending line changes to a descending line, are less obvious.  It would be nice to know more.
Can the t_i and alpha_i sequences be traced back to the start of the sequence? Of course the alpha_i sequence is not monotonic, and in particular we do not know at present if some alpha_i is equal to 5.
(End)
a(28149) = 7. - Chai Wah Wu, Oct 22 2023
Comment from N. J. A. Sloane, Mar 05 2024 (Start):
At present there is no OEIS entry for the inverse sequence, since it is not known if 5 appears here.
The initial values of the inverse sequence are
n.....1..2..3..4..5..6....7.....8..9..10..11... . . .
index.1..7..2..5..?..3..28149..81..?...?...4... . . . (End)

			For n = 2, prime(2-1) = prime(1) = 2; a(1) = 1, so a(1) mod 2 = 1, so a(2) is the least positive integer == 1 (mod 2) that has not yet appeared; 1 has appeared, so a(2) = 3.
For n = 3, prime(3-1) = 3; a(2) mod 3 = 0, so a(3) is the least unused integer == 0 mod 3, which is 6, so a(3) =  6.
For n = 4, prime(4-1) = 5; a(3) mod 5 = 1, and 6 has already been used, so a(4) = 11.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Sketch showing the main features of the graph
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, 2048 X 2048 raster showing a(n) , n = 1..4194304 in rows of 2048 terms, left to right, then continued below for 2048 rows total. Color indicates terms as follows: black = empty product {1}, red = prime (A40), gold = composite prime power (A246547), bright green = primorial A2110(k), k > 1, light green = squarefree semiprime (A6881 \ {6}), dark green = squarefree composite (A120944 \ {A2110 U A6881}), blue = numbers neither squarefree nor composite (A126706 \ A286708 = A332785), purple = squareful numbers that are not prime powers (A286708).
Chai Wah Wu, Graph of first 10^8 terms

Cf. A006509, A125717.
For a(n-1) (mod prime(n-1)) see A366470.
Records: A368384, A368385.
See also A366475, A366477.

a[1] = 1; a[n_] := a[n] = Module[{p = Prime[n - 1], k = 2, s = Array[a, n - 1]}, While[! FreeQ[s, k] || ! Divisible[k - a[n - 1], p], k++]; k]; Array[a, 100] (* Amiram Eldar, Oct 20 2023 *)
nn = 2^20; c[] := False; m[] := 0; a[1] = j = 1; c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}], n];
Array[a, nn] (* Michael De Vlieger, Oct 26 2023, fast, based on congruence, avoids search *)

from itertools import count, islice
from sympy import nextprime
def A364054_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    while True:
        yield a
        for b in count(a%p,p):
            if b not in aset:
                aset.add(b)
                a, p = b, nextprime(p)
                break
A364054_list = list(islice(A364054_gen(),30)) # Chai Wah Wu, Oct 22 2023

A099506 a(1)=1; for n > 1, a(n)=smallest m>0 that has not appeared so far in the sequence such that m+a(n-1) is a multiple of n.

Original entry on oeis.org

1, 3, 6, 2, 8, 4, 10, 14, 13, 7, 15, 9, 17, 11, 19, 29, 5, 31, 26, 34, 50, 16, 30, 18, 32, 20, 61, 23, 35, 25, 37, 27, 39, 63, 42, 66, 45, 69, 48, 72, 51, 33, 53, 79, 56, 36, 58, 38, 60, 40, 62, 94, 12, 96, 124, 44, 70, 46, 131, 49, 73, 113, 76, 52, 78, 54, 80, 192, 84, 126, 87

Offset: 1

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

			a(1)=1 by definition.
a(2)=3 because then a(2)+a(1)=3+1=4 which is a multiple of 2. a(2) cannot be 1 (which would lead to a sum of 2) because this has already appeared.
Likewise, a(3)=6 so that a(3)+a(2)=6+3=9 which is a multiple of 3.
a(4)=2 so that a(4)+a(3)=2+6=8 and so on.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A099507 for positions of occurrences of integers in this sequence.

Cf. A125717.

N = 100;
M = 10*N;  % find a(1) to a(N) or until a(n) > M
B = zeros(1,M);
A = zeros(1,N);
mmin = 2;
A(1) = 1;
B(1) = 1;
for n = 2:N
  for m = mmin:M
    if mmin == m && B(m) == 1
       mmin = mmin+1;
    elseif B(m) == 0 && rem(m + A(n-1),n) == 0
      A(n) = m;
      B(m) = 1;
      if m == mmin
         mmin = mmin + 1;
      end;
      break
    end;
  end;
  if A(n) == 0
     break
  end
end;
if A(n) == 0
  A(1:n-1)
else
  A
end; % Robert Israel, Jun 17 2015

v=[1];n=1;while(n<100,s=n+v[#v];if(!(s%(#v+1)||vecsearch(vecsort(v),n)),v=concat(v,n);n=0);n++);v \\ Derek Orr, Jun 16 2015

A367288 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(n-1) and a(n) are congruent modulo n, and the least value not yet in the sequence appears as soon as possible.

Original entry on oeis.org

0, 1, 5, 2, 18, 3, 39, 4, 60, 6, 106, 7, 151, 8, 204, 9, 265, 10, 334, 11, 411, 12, 496, 13, 589, 14, 690, 15, 799, 16, 916, 17, 1009, 19, 1175, 20, 1316, 21, 1465, 22, 1622, 23, 1787, 24, 1960, 25, 2141, 26, 2330, 27, 2527, 28, 2732, 29, 2945, 30, 3166, 31

Offset: 0

Rémy Sigrist, Nov 12 2023

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) and v are congruent modulo 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).
This construction is similar to that of A352713.
This sequence is a variant of A125717 and, by design, is guaranteed to be a permutation of the nonnegative integers (with inverse A367289).

			The first terms are:
  n   a(n)  a(n-1) mod n  a(n) mod n
  --  ----  ------------  ----------
   0     0  N/A           N/A
   1     1             0           0
   2     5             1           1
   3     2             2           2
   4    18             2           2
   5     3             3           3
   6    39             3           3
   7     4             4           4
   8    60             4           4
   9     6             6           6
  10   106             6           6
  11     7             7           7
  12   151             7           7
  13     8             8           8

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program
Index entries for sequences related to Recamán's sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A005132, A125717, A352713, A367289 (inverse), A367290.

PARI
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See Links section.
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A370956 Record high values in A245340.

Original entry on oeis.org

0, 1, 4, 8, 21, 1518, 2510, 4100, 11181, 18414, 30374, 50121, 82924, 136341, 611212, 4477981, 7351356, 12086260, 19861634, 32648059, 53646155, 144857355, 238062163, 643132294, 1736990151, 4691130396, 7709412048

Offset: 1

N. J. A. Sloane, Mar 13 2024

These numbers take a record number of steps to appear in A125717.

Cf. A125717, A245340, A245394, A245395, A370951.

See A370959 for the indices of these records in A245340.

from itertools import count, islice
def A370956_gen(): # generator of terms
    a, aset, b, c = 0, set(), 0, -1
    for n in count(1):
        aset.add(a)
        if a==b:
            if n-1>c:
                c = n-1
                yield c
            while b in aset:
                b += 1
        a = next(a for a in count(a%n,n) if a not in aset)
A370956_list = list(islice(A370956_gen(),20)) # Chai Wah Wu, Mar 28 2024

a(17)-a(24) from Michael S. Branicky, Mar 28 2024

a(25)-a(27) from Chai Wah Wu, Mar 28 2024

cp's OEIS Frontend

A245340 Smallest m such that A125717(m) = n, or -1 if n never appears.

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A245394 Record values in A125717.

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A245395 Where record values occur in A125717.

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A370957 First differences of A125717.

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A370958 Normalized first differences of A125717.

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Formula

A372057 After A125717(n) has been found, a(n) is the smallest missing number.

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A364054 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

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A099506 a(1)=1; for n > 1, a(n)=smallest m>0 that has not appeared so far in the sequence such that m+a(n-1) is a multiple of n.

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A367288 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(n-1) and a(n) are congruent modulo n, and the least value not yet in the sequence appears as soon as possible.

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A370956 Record high values in A245340.