A057176 -id:A057176 - cp's OEIS frontend

A087045 Indices k where A057176(k) = 1.

1, 4, 64, 400, 489, 519, 2164, 3589, 8703, 84761, 358837, 1463825, 1668392, 20471837, 31960443

Offset: 1

W. Edwin Clark and Farideh Firoozbakht at the suggestion of Leroy Quet, Aug 03 2003

By definition k is in the sequence if and only if k divides A057176(k-1). - Farideh Firoozbakht, Aug 04 2003

No more terms < 3870000. - David Wasserman, Mar 31 2005

			a(0) = 0 since A057176(0) = 1.

Cf. A057176.

N := 10^5: A := array(0..N): A[0] := 1; for m from 1 to N do A[m] := add(A[j],j=0..modp(A[m-1],m)): od: a := NULL: for i from 0 to N do if A[i]=1 then a := a,i; fi; od; a;

More terms from David Wasserman, Mar 31 2005

a(14)-a(15) from Michael S. Branicky, Mar 22 2024

A086809 Indices k where A057176(k) = 2.

Original entry on oeis.org

2, 5, 8, 34, 65, 144, 296, 401, 490, 520, 2165, 3590, 4640, 4828, 6828, 8704, 17675, 21164, 52883, 84762, 162069, 358838, 1463826, 1593474, 1668393, 2086706, 4364420, 20471838, 31960444

Offset: 1

Farideh Firoozbakht, Aug 05 2003

By definition of the sequence, A087045(n) + 1 for every n > 0 is in the sequence.

No more terms < 3870000. - David Wasserman, Mar 31 2005

Cf. A057176, A087045.

a[m_] := a[m] = Sum[ a[j], {j, 0, Mod[ a[m - 1], m]}]; vv = {}; Do[ a[n] = If[ n == 0, 1, b]; v = a[n + 1]; b = v; If[ v == 2, AppendTo[ vv, n + 1]; Print[n + 1]], {n, 0, 165000}]

Extended by Robert G. Wilson v, Aug 07 2003
More terms from David Wasserman, Mar 31 2005
a(27)-a(29) from Michael S. Branicky, Mar 23 2024

A086838 Indices k where A057176(k) = 4.

Original entry on oeis.org

3, 6, 9, 13, 35, 66, 115, 145, 297, 402, 491, 521, 1217, 2166, 3397, 3591, 4641, 4829, 6810, 6829, 7978, 8705, 17676, 21165, 50722, 52884, 84763, 121344, 121482, 162070, 358839, 560596, 1463827, 1593475, 1668394, 2086707, 2164032, 2517791, 4364421, 4970661, 19572673, 20471839, 31904004, 31960445, 51887007

Offset: 1

Farideh Firoozbakht, Aug 08 2003

nonn

By definition of the sequence A087045(n)+2 and A086809(n)+1 for every n > 0 are in the sequence. There are no further terms up to 200000

No more terms < 3870000. - David Wasserman, Mar 31 2005

Cf. A057176, A087045, A086809.

a[m_] := Sum[a[j], {j, 0, Mod[a[m-1], m]}]; vv={}; Do[ a[n]=If[n==0, 1, b]; v=a[n+1]; b=v; If[v==4, vv=Insert[vv, n+1, -1]; Print[vv]], {n, 0, 200000-1}]

More terms from David Wasserman, Mar 31 2005

a(39)-a(45) from Michael S. Branicky, Mar 23 2024

A357261 a(n) is the number of blocks in the bottom row after adding n blocks to the preceding structure of rows. See Comments and Example sections for more details.

Original entry on oeis.org

1, 3, 3, 3, 4, 1, 3, 1, 5, 4, 3, 3, 4, 6, 1, 3, 6, 3, 1, 7, 5, 3, 2, 2, 3, 5, 8, 1, 3, 6, 1, 6, 3, 1, 9, 6, 3, 1, 10, 7, 4, 2, 1, 1, 2, 4, 7, 11, 1, 3, 6, 10, 3, 9, 4, 12, 5, 11, 5, 13, 5, 11, 4, 12, 7, 3, 14, 8, 2, 12, 8, 5, 3, 2, 2, 3, 5

Offset: 1

John Tyler Rascoe, Oct 08 2022

A structure of rows is built up successively where each n blocks are added onto the preceding structure. The first row has an initial width of 3. After n = 1, n blocks are first added filling in the last row where n - 1 left off.
Once a row is filled a new row is started below it. After adding n blocks, if the final row reached is filled exactly, then the width of the next row is increased by one. Otherwise the width of the next row is the same as the previous.
Assuming the rows are built according to the given algorithm, a(n) is the number of blocks tagged 'n' in the last row that includes a block tagged 'n'." - Peter Luschny, Dec 20 2022
Will this sequence ever reach a point after which a(n) grows linearly? This is the case using an initial width of 2 instead of 3.

			After blocks 1 and 2, the initial row width 3 is exactly filled and hence the next row (of 3's and 4) is 1 longer.
  |1|2|2|       initial row
  |3|3|3|4|
  |4|4|4|5|
  |5|5|5|5|
  |6|6|6|6|6|
  |6|_|_|_|_|   last row after n=6
For n=6, the structure after blocks 1 through 6 have been added is as shown above and its final row has just one block (one 6) so that a(6) = 1.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A002024, A057176, A064434, A096535, A104647, A275204.

A357261_list := proc(maxn) local A, g, c, n, r;
A := []; g := 3; c := 0;
for n from 1 to maxn do
    r := irem(n + c, g);
    c := r;
    if r = 0 then
        r := g; g := g + 1;
    fi;
    A := [op(A), r];
od; return A end:
A357261_list(77); # Peter Luschny, Dec 21 2022

lista(nn) = my(nbc=3, col=0, list=List()); for (n=1, nn, col = lift(Mod(col+n, nbc)); listput(list, if (col, col, nbc)); if (!col, nbc++);); Vec(list); \\ Michel Marcus, Oct 17 2022

def A357261_list(maxn):
    """Returns a list of the first maxn terms"""
    A = []
    g = 3
    c = 0
    for n in range(1,maxn+1):
        if (n + c)%g == 0:
            A.append(g)
            g += 1
            c = 0
        else:
            A.append((n + c)%g)
            c = A[-1]
    return A

A130618 a(1)=1; a(n+1) = Sum_{k=0..a(n) mod n} a(n-k).

Original entry on oeis.org

1, 1, 2, 4, 4, 12, 12, 35, 63, 63, 173, 368, 734, 1448, 2884, 5607, 11340, 16947, 39627, 79301, 118928, 271750, 543500, 1092066, 2184858, 4358317, 8727848, 17455759, 34911652, 61095259, 130918366, 244381036, 506138640, 1012353685, 2024551664

Offset: 1

Leroy Quet, Jun 18 2007

nonn

			a(10) mod 10 = 63 mod 10 = 3. So a(11) = Sum_{k=0..3} a(10-k) = a(10) + a(9) + a(8) + a(7) = 63 + 63 + 35 + 12 = 173.

Vaclav Kotesovec, Table of n, a(n) for n = 1..3300
Vaclav Kotesovec, Plot of a(n) / (2^n/n) for n = 1..50000

Cf. A057176.

a[1] := 1; for n to 35 do a[n+1] := add(a[n-k], k = 0 .. `mod`(a[n], n)) end do; seq(a[n], n = 1 .. 35); # Emeric Deutsch, Jun 21 2007

a[1] = 1; a[n_] := a[n] = Sum[a[n-1-k], {k, 0, Mod[a[n-1], n-1]}]; Table[a[n], {n, 1, 50}] (* Vaclav Kotesovec, Apr 26 2020 *)

More terms from Jon E. Schoenfield, Jun 21 2007

More terms from Emeric Deutsch, Jun 21 2007

A358073 a(n) is the row position of the n-th number n after adding the number n, n times to the preceding triangle. A variant of A357261, see Comments and Examples for more details.

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 4, 3, 3, 4, 6, 9, 13, 6, 21, 16, 33, 15, 34, 18, 3, 25, 12, 36, 25, 51, 18, 46, 15, 45, 16, 48, 21, 55, 30, 6, 43, 21, 60, 40, 81, 24, 67, 12, 57, 4, 51, 99, 49, 99, 3, 55, 108, 15, 70, 126, 36, 94, 6, 66, 127, 42, 105, 22, 87, 6, 73, 141, 63

Offset: 1

John Tyler Rascoe, Oct 29 2022

A triangle is built up successively where n appears n times within the triangle. Each row has a set width before n is added, and the first row begins with a width of 1.
Numbers n are added to the first open position within the triangle or where the previous n left off so that no gaps are left in the rows of the triangle. If the row position of the n-th number n placed is the rightmost position within that row, then the width of the next row is increased by n. Otherwise, the width of the next row stays the same as the previous one.
The next row's width can only increase after a given n is added all n times. So when a row is filled after adding fewer than n n's, the next row, by definition, will have the same width.

			After 5 is added 5 times, the fifth 5 falls in the rightmost row position. So the width of the next row is increased by 5.
  |1|       initial row
  |2|2|
  |3|3|3|4|
  |4|4|4|5|
  |5|5|5|5|
  |6|6|6|6|6|6|7|7|7|
  |7|7|7|7|_|_|_|_|_|
a(7) = 4 because the row position of the seventh 7 added is 4.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
John Tyler Rascoe, Scatterplot of a(n) for n = 1...50000

Cf. A002024, A057176, A064434, A096535, A104647, A275204, A357261.

A358073_list := proc(maxn)  local A, g, c, n, r;
A := []; g := 1; c := 0;
for n from 1 to maxn do
    r := irem(n + c, g);
    c := r;
    if r = 0 then
        r := g;
        g := g + n;
    fi;
    A := [op(A), r];
od; return A end:
A358073_list(69); # Peter Luschny, Dec 21 2022

def A358073_list(maxn):
    """Returns a list of the first maxn terms"""
    A = []
    g = 1
    c = 0
    for n in range(1,maxn+1):
        if (n + c)%g ==0:
            A.append(g)
            g += n
            c = 0
        else:
            A.append((n + c)%g)
            c = A[-1]
    return A

cp's OEIS Frontend

A087045 Indices k where A057176(k) = 1.

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A086809 Indices k where A057176(k) = 2.

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A086838 Indices k where A057176(k) = 4.

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A357261 a(n) is the number of blocks in the bottom row after adding n blocks to the preceding structure of rows. See Comments and Example sections for more details.

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A130618 a(1)=1; a(n+1) = Sum_{k=0..a(n) mod n} a(n-k).

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A358073 a(n) is the row position of the n-th number n after adding the number n, n times to the preceding triangle. A variant of A357261, see Comments and Examples for more details.

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