A006509 -id:A006509 - cp's OEIS frontend

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)).

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

A093903 a(1) = 1; for n > 1, a(n) = a(n-1)-p if there exists a prime p (take the smallest) that has not yet been used and is such that a(n) is new and > 0, otherwise a(n) = a(n-1)+p if the same conditions are satisfied.

Original entry on oeis.org

1, 3, 6, 11, 4, 15, 2, 19, 38, 9, 32, 63, 26, 67, 24, 71, 18, 77, 16, 83, 12, 85, 164, 81, 170, 73, 174, 65, 168, 61, 188, 75, 206, 69, 208, 59, 210, 53, 216, 49, 222, 43, 224, 33, 226, 29, 228, 17, 240, 13, 242, 475, 236, 477, 220, 471, 202, 465, 194, 487, 204, 481, 200

Offset: 1

Amarnath Murthy, May 24 2004

A variation of Cald's sequence A006509; a sequence of distinct positive integers with property that absolute successive differences are distinct primes.
A more long-winded definition: Start with a(1) = 1. Keep a list of the primes that have been used so far; initially this list is empty. Each prime can be used at most once.
To get a(n), subtract from a(n-1) each prime p < a(n-1) that has not yet been used, starting from the smallest. If for any such p, a(n-1)-p is not yet in the sequence, set a(n) = a(n-1)-p and mark p as used.
If no p works, then add each prime p that has not yet been used to a(n-1), again starting with the smallest. When p is such that a(n-1)+p is not yet in the sequence, set a(n) = a(n-1)+p and mark p as used. Repeat.
The main question is: does every number appear in the sequence?

			1 -> 1+2 = 3 and prime 2 has been used.
3 -> 3+3 = 6 and prime 3 has been used.
6 could go to 6-5 = 1, except 1 is already in the sequence; so 6 -> 6+5 = 11 and prime 5 has been used.
11 -> 11-7 = 4 (for the first time we can subtract) and prime 7 has been used.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Similar to Cald's sequence A006509 and Recamán's sequence A005132. Differs from A006509. Cf. A094746 (the primes associated with this sequence), A113959 (where n appears), A113960, A113961, A113962.

Cf. A000040.

import Data.List (delete)
a093903 n = a093903_list !! (n-1)
a093903_list = 1 : f [1] a000040_list where
   f xs@(x:_) ps = g ps where
     g (q:qs) | x <= q         = h ps
              | y `notElem` xs = y : f (y:xs) (delete q ps)
              | otherwise      = g qs where
       y = x - q
       h (r:rs) | z `notElem` xs = z : f (z:xs) (delete r ps)
                | otherwise      = h rs where
         z = x + r
-- Reinhard Zumkeller, Oct 17 2011

Definition (and sequence) corrected by R. Piyo (nagoya314(AT)yahoo.com) and N. J. A. Sloane, Dec 09 2004

Edited, offset changed to 1, a(16) and following terms added by Klaus Brockhaus, Nov 10 2005

A100707 a(1) = 1; for n > 1, a(n+1)=a(n)-k if there exists a positive number k (take the smallest) that has not yet been used and is such that a(n+1) is new and >0, otherwise a(n+1) = a(n)+k if the same conditions are satisfied.

Original entry on oeis.org

1, 2, 4, 7, 3, 8, 14, 6, 13, 22, 12, 23, 11, 24, 10, 25, 9, 26, 5, 27, 45, 21, 40, 20, 43, 18, 44, 17, 46, 16, 47, 19, 51, 15, 48, 82, 42, 77, 39, 76, 37, 78, 36, 79, 35, 80, 34, 81, 33, 83, 32, 84, 31, 85, 30, 86, 29, 87, 38, 97, 28, 88, 149, 75, 137, 74

Offset: 1

N. J. A. Sloane and Vinay Vaishampayan, Dec 10 2004

A sequence of distinct natural numbers with the property that absolute successive differences are distinct.
A more long-winded definition: start with a(1) = 1. We keep a list of the numbers k that have been used as differences so far; initially this list is empty. Each difference can be used at most once.
Suppose a(n) = M. To get a(n+1), we subtract from M each number k < M that has not yet been used, starting from the smallest. If for any such k, M-k is a number not yet in the sequence, set a(n+1) = M-k and mark the difference k as used.
If no k works, then we add each number k that has not yet been used to M, again starting with the smallest. When we find a k such that M+k is a number not yet in the sequence, we set a(n+1) = M+k and mark k as used. Repeat.
The main question is: does every number appear in the sequence?
A227617(n) = smallest m such that a(m) = n: if this sequence is a permutation of the natural numbers, then A227617 is its inverse. - Reinhard Zumkeller, Jul 19 2013

			1 -> 1+1 = 2 and k=1 has been used as a difference.
2 -> 2+4 = 4 and k=2 has been used as a difference.
4 could go to 4-3 = 1, except that 1 has already appeared in the sequence; so 4 -> 4+3 = 7 and k=3 has been used as a difference.
7 -> 7-4 = 3 (for the first time we can subtract) and k=4 has been used as a difference. And so on.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Similar to Murthy's sequence A093903, Cald's sequence (A006509) and Recamán's sequence A005132. See also A081145, A100709 (another version). Cf. A100708 (the successive differences associated with this sequence).

import Data.List (delete)
import qualified Data.Set as Set (insert)
import Data.Set (singleton, member)
a100707 n = a100707_list !! (n-1)
a100707_list = 1 : f 1 (singleton 1) [1..] where
   f y st ds = g ds where
     g (k:ks) | v <= 0      = h ds
              | member v st = g ks
              | otherwise   = v : f v (Set.insert v st) (delete k ds)
              where v = y - k
     h (k:ks) | member w st = h ks
              | otherwise   = w : f w (Set.insert w st) (delete k ds)
              where w = y + k
-- Reinhard Zumkeller, Jul 19 2013

Data corrected for n > 46 by Reinhard Zumkeller, Jul 19 2013

A110080 a(1) = 1; skipping over integers occurring earlier in the sequence, count down p(n) (p(n) = n-th prime) from a(n) to get a(n+1). If this is <= 0, instead count up from a(n) p(n) positions (skipping already occurring integers) to get a(n+1).

Original entry on oeis.org

1, 3, 6, 11, 2, 16, 29, 10, 32, 4, 39, 70, 31, 75, 27, 80, 20, 87, 17, 94, 9, 97, 176, 91, 183, 81, 188, 77, 193, 73, 198, 57, 203, 50, 206, 38, 209, 28, 216, 22, 223, 12, 226, 417, 222, 422, 219, 435, 202, 440, 199, 445, 190, 448, 177, 455, 169, 462, 166, 469, 161, 472

Offset: 1

Leroy Quet, Oct 12 2005

If we did not skip earlier occurring integers when counting, we would instead have Cald's sequence (A006509).

			The first 5 terms of the sequence can be plotted on the number line as:
1,2,3,*,*,6,*,*,*,*,11,*,*,*,*,*.
a(5) is 2. Counting p(5) = 11 down from 2 gets a negative integer. So we instead count up 11 positions, skipping the 3, 6 and 11 as we count, to arrive at 16 (which is at the rightmost * of the number line above).
Here is the calculation of the first 6 terms in more detail:
integers i : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
i at n = ... : 1 5 2 . . 3 . . . .. .4 .. .. .. .. .6 ...
prime p used : - 7 2 . . 3 . . . .. .5 .. .. .. .. 11 ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A091023, A091263, A006509, A111187 (inverse).

import Data.Set (singleton, member, insert)
a110080 n = a110080_list !! (n-1)
a110080_list = 1 : f 1 a000040_list (singleton 1) where
   f x (p:ps) m = y : f y ps (insert y m) where
     y = g x p
     g 0 _ = h x p
     g u 0 = u
     g u v = g (u - 1) (if member (u - 1) m then v else v - 1)
     h u 0 = u
     h u v = h (u + 1) (if member (u + 1) m then v else v - 1)
-- Reinhard Zumkeller, Sep 02 2014

More terms from Klaus Brockhaus and Hans Havermann, Oct 17 2005

A117128 Recamán transform of primes (another version): a(0)=1; for n>0, a(n) = a(n-1) - prime(n) if that number is positive and not already in the sequence, otherwise a(n) = a(n-1) + prime(n).

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, 275, 162, 35, 166, 29, 168, 317, 468, 311, 148, 315, 142, 321, 140, 331, 138, 335, 136, 347, 124, 351, 122, 355, 116, 357, 106, 363, 100, 369, 98, 375, 94, 377

Offset: 0

N. J. A. Sloane, Apr 20 2006

nonn

Differs from Cald's sequence A006509 for first time at n=116 (or 117, depending on offset).

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

Cf. A064365, A006509, A112877.

import Data.Set (singleton, notMember, insert)
a117128 n = a117128_list !! n
a117128_list = 1 : f 1 a000040_list (singleton 1) where
   f x (p:ps) s | x' > 0 && x' `notMember` s = x' : f x' ps (insert x' s)
                | otherwise                  = xp : f xp ps (insert xp s)
                where x' = x - p; xp = x + p
-- Reinhard Zumkeller, Apr 26 2012

M1:=500000; a:=array(0..M1); have:=array(1..M1); a[0]:=1; for n from 1 to M1 do have[n]:=0; od: have[1]:=1;
M2:=2000; nmax:=M2;
for n from 1 to M2 do p:=ithprime(n); i:=a[n-1]-p; j:=a[n-1]+p;
if i >= 1 and have[i]=0 then a[n]:=i; have[i]:=1;
elif j <= M1 then a[n]:=j; have[j]:=1;
else nmax:=n-1; break; fi; od: [seq(a[n],n=0..M2)];

a = {1}; Do[If[And[#1 > 0, ! MemberQ[a, #1]], AppendTo[a, #1], AppendTo[a, #2]] & @@ {#1 - #2, #1 + #2} & @@ {a[[n - 1]], Prime[n - 1]}, {n, 2, 62}]; a (* Michael De Vlieger, Dec 05 2016 *)

from sympy import primerange, prime
def aupton(terms):
  alst = [1]
  for n, pn in enumerate(primerange(1, prime(terms)+1), start=1):
    x = alst[-1] - pn
    alst += [x if x > 0 and x not in alst else alst[-1] + pn]
  return alst
print(aupton(61)) # Michael S. Branicky, May 30 2021

a(n) = A064365(n) + 1. - Thomas Ordowski, Dec 05 2016

A368382 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 A004280(n)).

Original entry on oeis.org

1, 3, 6, 11, 4, 13, 2, 15, 30, 47, 9, 51, 5, 55, 28, 57, 26, 59, 24, 61, 22, 63, 20, 65, 18, 67, 16, 69, 14, 71, 12, 73, 10, 75, 8, 77, 148, 221, 146, 223, 144, 225, 142, 227, 53, 231, 49, 235, 45, 239, 41, 243, 37, 247, 33, 251, 29, 255, 25, 259, 21, 263, 17, 267, 140, 269, 7, 273, 138, 275, 136, 277, 134, 279, 132, 281

Offset: 1

N. J. A. Sloane, Mar 03 2024

nonn

Analogous to A364054, but whereas that sequence is based on the sequence of primes (2, 3, 5, 7, 11, ....), the present sequence is based on the sequence 2, 3, 5, 7, 9, 11, 13, 15, ... (2 together with the odd numbers >1, essentially A004280).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A004280.

Similar definitions: A005132, A006509, A364054.

from itertools import count, islice
def A368382_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, 3 if p == 2 else p+2
                break
A368382_list = list(islice(A368382_gen(),40)) # Chai Wah Wu, Mar 05 2024

A094746 Primes arising as successive differences in A093903.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 109, 103, 107, 127, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 257, 251, 269, 263, 271, 293

Offset: 1

Amarnath Murthy, May 24 2004

nonn

Differs from A006509.

Cf. A093903, A006509, A113962.

Absolute(A093903(n+1)-A093903(n))

Corrected by R. Piyo (nagoya314(AT)yahoo.com) and N. J. A. Sloane, Dec 09 2004

a(15) and following terms from Klaus Brockhaus, Nov 10 2005

A112878 A100298(n) = 0.

Original entry on oeis.org

14, 34, 35, 41, 42, 56, 92, 100, 124, 132, 133, 134, 135, 153, 167, 204, 216, 217, 218, 236, 261, 269, 283, 295, 330, 342, 369, 373, 392, 443, 464, 480, 488, 521, 587, 599, 634, 648, 688, 714, 723, 742, 786, 814, 815, 816, 817, 818, 819, 820, 821, 829, 833

Offset: 1

Klaus Brockhaus, Oct 24 2005

nonn

A100298 is similar to Cald's sequence A006509, where zeros are much less frequent (cf. A112877).

			A100298(14) = 0 and A100298(k) > 0 for k < 14, so a(1) = 14.

Cf. A100298, A006509, A112877.

A367188 a(1) = 1, thereafter a(n) = a(n-1) - A007504(n-1) if positive and novel, else a(n-1) + A007504(n-1).

Original entry on oeis.org

1, 3, 8, 18, 35, 7, 48, 106, 29, 129, 258, 98, 295, 57, 338, 10, 391, 831, 330, 898, 259, 971, 180, 1054, 91, 1151, 2312, 1048, 2419, 939, 2532, 812, 2663, 675, 2802, 526, 2953, 369, 3116, 202, 3289, 23, 3470, 7108, 3277, 7305, 3078, 7516, 2855, 7743, 2626, 7976

Offset: 1

David James Sycamore, Nov 10 2023

nonn

A variation on Cald's sequence A006509.

			a(1)-A007504(1) = 1-2, negative so a(2) = 1+2 = 3.
a(2)-A007504(2) = 3-5, negative so a(3) = 3+5 = 8.
a(3)-A007504(3) = 8-10, negative so a(4) = 8+10 =18.
a(4)-A007504(4) = 18-17 = 1 (positive but seen before at a(1)), so a(5) = 35.
a(5)-A007504(5) = 35-28 = 7, positive and novel so a(6) = 7.
a(10)-A007504(10) = 129-129 = 0, therefore a(11) = 2*129 = 258.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A000040, A006509, A007504.

nn = 120; c[_] := False; a[1] = j = 1; c[1] = True; s = 2;
Do[If[Or[# < 1, c[#]], Set[k, j + s], Set[k, #]] &[j - s];
  s += Prime[n];
  Set[{a[n], j, c[k]}, {k, k, True}], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Nov 10 2023 *)

More terms from Michael De Vlieger, Nov 10 2023

cp's OEIS Frontend

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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A093903 a(1) = 1; for n > 1, a(n) = a(n-1)-p if there exists a prime p (take the smallest) that has not yet been used and is such that a(n) is new and > 0, otherwise a(n) = a(n-1)+p if the same conditions are satisfied.

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A100707 a(1) = 1; for n > 1, a(n+1)=a(n)-k if there exists a positive number k (take the smallest) that has not yet been used and is such that a(n+1) is new and >0, otherwise a(n+1) = a(n)+k if the same conditions are satisfied.

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A110080 a(1) = 1; skipping over integers occurring earlier in the sequence, count down p(n) (p(n) = n-th prime) from a(n) to get a(n+1). If this is <= 0, instead count up from a(n) p(n) positions (skipping already occurring integers) to get a(n+1).

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A117128 Recamán transform of primes (another version): a(0)=1; for n>0, a(n) = a(n-1) - prime(n) if that number is positive and not already in the sequence, otherwise a(n) = a(n-1) + prime(n).

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A368382 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 A004280(n)).

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A094746 Primes arising as successive differences in A093903.

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A112878 A100298(n) = 0.

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