A340612 -id:A340612 - cp's OEIS frontend

A340593 a(0) = 0; for n > 0, if n appears in the sequence then a(n) = a(n-1) - lastindex(n) if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

0, 1, 3, 5, 9, 6, 11, 4, 12, 8, 18, 24, 16, 29, 15, 29, 17, 33, 23, 42, 22, 43, 63, 45, 34, 59, 85, 58, 30, 45, 73, 104, 72, 55, 31, 66, 102, 65, 27, 66, 26, 67, 48, 69, 25, 54, 100, 53, 95, 46, 96, 147, 199, 152, 107, 74, 130, 187, 160, 135, 75, 14, 76, 98, 162, 125, 86, 127, 195, 238, 168, 97

Offset: 0

Scott R. Shannon, Jan 13 2021

nonn

This sequences uses the same rules as Recamán's sequence A005132 if the value of n itself has not previously appeared in the sequence. However if n has previously appeared then the step size from a(n-1) is set to lastindex(n), where lastindex(n) is the sequence index of the last appearance of n.

The smallest value not to have appeared after 1 million terms is 52. It is unknown if all terms eventually appear.

			a(3) = 5 as a(2) = 3 = n, thus the step size from a(2) is 2. As 1 has previously appeared a(3) = a(2) + 2 = 3 + 2 = 5.
a(5) = 6 as a(3) = 5 = n, thus the step size from a(4) is 3. As 6 has not previously appeared a(5) = a(4) - 3 = 9 - 3 = 6.

Cf. A005132, A339673, A340612, A336760, A336761.

from itertools import count, islice
def A340593_gen(): # generator of terms
    a, ndict = 0, {0:0}
    yield 0
    for n in count(1):
        yield (a:= (a-m if a>=(m:=ndict[n]) and a-m not in ndict else a+m) if n in ndict else (a-n if a>=n and a-n not in ndict else a+n))
        ndict[a] = n
A340593_list = list(islice(A340593_gen(),30)) # Chai Wah Wu, Jun 29 2023

A362332 For n > 1, if n appears in the sequence then a(n) = lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)(n+1), a(1) = 1 and a(2) = 12.

Original entry on oeis.org

1, 2, 6, 24, 120, 3, 21, 168, 1512, 15120, 166320, 13860, 180180, 12870, 858, 13728, 233376, 4200768, 79814592, 1596291840, 7, 154, 3542, 4, 100, 2600, 70200, 1965600, 57002400, 1900080, 58902480, 1884879360, 62201018880

Offset: 1

Kelvin Voskuijl, Apr 16 2023

nonn

			a(2) = 2, as a(1) = 1 and 1 times 2 is 2.
a(6) = 3, as a(3) = 6 = n, thus a(6) = 3.
a(7) = 21, as a(6) = 3 and 3 times 7 is 21.

Michael De Vlieger, Table of n, a(n) for n = 1..4306
Michael De Vlieger, Plot p(i)^e(i) | a(n) at (x,y) = (n,i) for n = 1..2048, with a color function representing e(i), where black represents e(i) = 1, red e(i) = 2, etc., 3X vertical exaggeration.

Cf. A008336, A340612.

nn = 33; c[] := 0; Array[Set[{a[#], c[#]}, {#, #}] &, 2]; j = 2; Do[If[c[n] > 0, Set[k, c[n]], If[Divisible[j, n], Set[k, j/n], Set[k, j*n]]]; Set[{a[n], c[k], j}, {k, n, k}], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Jun 23 2023 *)

pos(n, va) = for (k=1, #va, if (va[k] == n, return (k)););
lista(nn) = my(va = vector(nn)); for (n=1, 2, va[n] = n); for (n=3, nn, my(p=pos(n, va)); if (p, va[n] = p, if (va[n-1] % n, va[n] = n*va[n-1], va[n] = va[n-1]/n); ); ); va; \\ Michel Marcus, Jun 23 2023

from itertools import count, islice
def A362332_gen(): # generator of terms
    a, ndict = 2, {1:1,2:2}
    yield from [1,2]
    for n in count(3):
        yield (a:= ndict[n] if n in ndict else (a*n if a%n else a//n))
        ndict[a] = n
A362332_list = list(islice(A362332_gen(),30)) # Chai Wah Wu, Jun 29 2023

A362373 a(0) = 0; for n > 0, if n appears in the sequence then a(n) is the sum of the indices of all previous appearances of n. Otherwise a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

0, 1, 3, 2, 6, 11, 4, 11, 19, 10, 9, 12, 11, 24, 38, 23, 7, 24, 42, 8, 28, 49, 27, 15, 30, 5, 31, 22, 20, 49, 24, 26, 58, 25, 59, 94, 130, 93, 14, 53, 13, 54, 18, 61, 17, 62, 16, 63, 111, 50, 49, 100, 48, 39, 41, 96, 40, 97, 32, 34, 94

Offset: 0

Kelvin Voskuijl, Apr 17 2023

Conjecture: All nonnegative integers appear in this sequence. - Yifan Xie, Apr 24 2023

			a(2) = 3 = n, thus a(3) = 2.
a(5) = 11, as 5 has not previously appeared in the sequence, but a(4) - 5 = 1 has, thus a(5) = a(4) + 5 = 6 + 5 = 11.
a(5) and a(7) = 11, and 5 + 7 = 12, thus a(11) = 12.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20

Cf. A005132, A340612.

nn = 120; c[] := 0; j = a[0] = 0; Do[If[# > 0, Set[k, #], If[And[n <= j, c[#] == 0], Set[k, #], Set[k, j + n]] &[j - n] ] &[c[n]]; c[k] += n; Set[{a[n], j}, {k, k}], {n, nn}]; Array[a, nn, 0] (* _Michael De Vlieger, Apr 19 2023 *)

from itertools import count, islice
def A362373_gen(): # generator of terms
    a, ndict = 0, {0:0}
    yield 0
    for n in count(1):
        yield (a:= ndict[n] if n in ndict else (a-n if a>=n and a-n not in ndict else a+n))
        ndict[a] = ndict.get(a,0)+n
A363373_list = list(islice(A362373_gen(),30)) # Chai Wah Wu, Jun 29 2023

A340733 a(0) = 0; for n > 0, if the value of n itself appears in the sequence then a(n) = a(n-1) - (n-index(n)) if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + (n-index(n)), where index(n) is the index of the last appearance of n. If the value of n does not appear then a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

0, 1, 3, 2, 6, 11, 9, 16, 8, 5, 15, 21, 33, 20, 34, 29, 38, 55, 37, 18, 25, 35, 13, 36, 12, 7, 33, 60, 32, 46, 76, 45, 41, 48, 28, 14, 27, 46, 24, 63, 23, 32, 74, 31, 75, 61, 52, 99, 84, 133, 83, 134, 128, 181, 127, 89, 145, 88, 30, 89, 56, 40, 102, 78, 142, 77, 143, 210, 278, 209, 139, 68

Offset: 0

Scott R. Shannon, Jan 17 2021

nonn

This sequence uses the same rules as Recamán's sequence A005132 if the value of n itself has not previously appeared in the sequence. However if n has previously appeared then the step size from a(n-1) is set to be n - index(n), where index(n) is the sequence index of the last appearance of n.

The sequence values appear random up to a(45256) = 45257. As 45257 has then appeared in the sequence the step size for the next term is 45257 - index(45257) = 45257 - 452576 = 1. As a(42564) = 45256 the next term a(45257) must be a(45256) + 1 = 45257 + 1 = 45258. This pattern then repeats so all terms beyond a(45256) are just a(n-1) + 1. See the linked image.

			a(3) = 2 as a(2) = 3 = n, thus the step size from a(2) is 3 - index(3) = 3 - 2 = 1. As 2 has not previously appeared a(3) = a(2) - 1 = 3 - 1 = 2.
a(6) = 9 as a(4) = 6 = n, thus the step size from a(5) is 6 - index(6) = 6 - 4 = 2. As 9 has not previously appeared a(6) = a(5) - 2 = 11 - 2 = 9.

Scott R. Shannon, Image of the first 60000 terms.

Cf. A005132, A340593, A340612, A336760, A336761.

from itertools import count, islice
def A340733_gen(): # generator of terms
    a, ndict = 0, {0:0}
    yield 0
    for n in count(1):
        yield (a:= (a-m if a>=(m:=n-ndict[n]) and a-m not in ndict else a+m) if n in ndict else (a-n if a>=n and a-n not in ndict else a+n))
        ndict[a] = n
A340733_list = list(islice(A340733_gen(),30)) # Chai Wah Wu, Jun 29 2023

a(n) = n + 1 for n >= 45256.

A362698 For n > 1, if n appears in the sequence, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)(n+1), a(1) = 1 and a(2) = 12.

Original entry on oeis.org

1, 2, 6, 24, 120, 114, 798, 6384, 57456, 574560, 6320160, 526680, 6846840, 489060, 32604, 521664, 8868288, 159629184, 8401536, 168030720, 3528645120, 160392960, 3689038080, 3689038056, 92225951400, 2397874736400, 64742617882800, 1812793300718400, 52571005720833600

Offset: 1

Kelvin Voskuijl, Jul 07 2023

nonn

			a(2) = 2, as a(1) = 1 and 1 times 2 is 2.
a(6) = 114, as a(3) = 6 = n, thus a(6) = a(5) - 6 =  114.
a(7) = 798, as a(6) = 114 and 7 times 114 is 798.

Michael De Vlieger, Table of n, a(n) for n = 1..1996
Michael De Vlieger, Log log scatterplot of log_10 a(n), n = 2..2^18.

Cf. A008336, A362332, A340612.

nn = 120; c[] := False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; Do[k = If[c[n], If[And[! c[#], # >= 0], #, j + n] &[j - n], If[Divisible[j, n], j/n, j n]]; Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Aug 30 2023 *)

More terms from Michael De Vlieger, Aug 30 2023

A363962 a(1)=1; for n > 1, if n appears in the sequence then a(n) = lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n) = a(n-1) - n unless that result is already in the sequence or would be negative; otherwise, a(n) = a(n-1) * n.

Original entry on oeis.org

1, 2, 6, 24, 19, 3, 21, 13, 4, 40, 29, 17, 8, 112, 97, 81, 12, 216, 5, 100, 7, 154, 131, 4, 100, 74, 47, 1316, 11, 330, 299, 267, 234, 200, 165, 129, 92, 54, 15, 10, 410, 368, 325, 281, 236, 190, 27, 1296, 1247, 1197, 1146, 1094, 1041, 38, 2090, 2034, 1977, 1919, 1860, 1800, 1739, 1677, 1614

Offset: 1

Kelvin Voskuijl, Jun 29 2023

nonn

			a(6) = 3, as a(3) = 6 = n, thus a(6) = 3.
a(28) = 1316 because subtracting 28 from the previous term (47) would give 19, which is already in the sequence, so multiply 47 by 28 to get 1316.
a(100) = 25, as a(25) = 100 = n, thus a(100) = 25.

Cf. A005132, A340612, A354833, A362332.

function a = A363962( max_n )
    a = 1;
    for n = 2:max_n
        r = find(a == n,1,'last');
        if ~isempty(r)
            a(n) = r;
        else
            k = a(n-1) - n;
            if k > 0 && isempty(find(a == k, 1))
                a(n) = k;
            else
                a(n) = a(n-1) * n;
            end
        end
    end
end % Thomas Scheuerle, Jul 03 2023

a[1] = 1; a[n_] := a[n] = Module[{s = Array[a, n - 1], a1}, If[MemberQ[s, n], Position[s, n][[-1, 1]], a1 = a[n - 1] - n; If[a1 < 0 || MemberQ[s, a1], a[n - 1]*n, a1]]]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

A364929 a(0) = 0; for n > 0, if n appears in the sequence then a(n) is the product of the indices of all previous appearances of n. Otherwise a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

0, 1, 3, 2, 6, 11, 4, 11, 19, 10, 9, 35, 23, 36, 22, 7, 23, 40, 58, 8, 28, 49, 14, 192, 168, 143, 117, 90, 20, 49, 79, 48, 16, 49, 15, 11, 13, 50, 12, 51, 17, 58, 100, 57, 101, 56, 102, 55, 31, 20097, 37, 39, 91, 38, 92, 47, 45, 43, 738, 679, 619, 558, 496, 433, 369

Offset: 0

Kelvin Voskuijl, Dec 05 2023

nonn

			a(2) = 3 = n, thus a(3) = 2.
a(5) = 11, as 5 has not previously appeared in the sequence, but a(4) - 5 = 1 has, thus a(5) = a(4) + 5 = 6 + 5 = 11.
a(5) and a(7) = 11, and 5 * 7 = 35, thus a(11) = 35.

Michael S. Branicky, Table of n, a(n) for n = 0..10000
David A. Corneth, PARI program

Cf. A005132, A340612, A362373.

PARI
```
See PARI link
```

from itertools import count, islice
def agen(): # generator of terms
    an, p = 0, {0: 0} # data list, map of product of indices
    for n in count(1):
        yield an
        an = p[n] if n in p else an+n if an-n < 0 or an-n in p else an-n
        p[an] = p[an]*n if an in p else n
print(list(islice(agen(), 65))) # Michael S. Branicky, Dec 09 2023

More terms from David A. Corneth, Dec 09 2023

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A362332 For n > 1, if n appears in the sequence then a(n) = lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)*(n+1), a(1) = 1 and a(2) = 1*2.

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A362373 a(0) = 0; for n > 0, if n appears in the sequence then a(n) is the sum of the indices of all previous appearances of n. Otherwise a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

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A362698 For n > 1, if n appears in the sequence, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)*(n+1), a(1) = 1 and a(2) = 1*2.

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A363962 a(1)=1; for n > 1, if n appears in the sequence then a(n) = lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n) = a(n-1) - n unless that result is already in the sequence or would be negative; otherwise, a(n) = a(n-1) * n.

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A364929 a(0) = 0; for n > 0, if n appears in the sequence then a(n) is the product of the indices of all previous appearances of n. Otherwise a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

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A362332 For n > 1, if n appears in the sequence then a(n) = lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)(n+1), a(1) = 1 and a(2) = 12.

A362698 For n > 1, if n appears in the sequence, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)(n+1), a(1) = 1 and a(2) = 12.