A229037 -id:A229037 - cp's OEIS frontend

A381598 Index of first term of three consecutive n's in A381597.

1, 9, 34, 147, 111, 359, 437, 389, 594, 826, 1102, 83317, 1789, 5142, 2931, 12671

Offset: 1

Scott R. Shannon, Mar 01 2025

The terms vary greatly in size - after 5.2 million terms of A381597 no three consecutive 17's or 18's have appeared, although three consecutive 19's appear at index 6474. The largest known term is a(192) = 5135798.

Cf. A381597, A381599, A370708, A281511, A229037.

A381599 Index where n first appears in A381597.

Original entry on oeis.org

1, 4, 17, 39, 68, 124, 191, 286, 441, 577, 776, 1043, 1192, 1556, 1736, 2214, 2744, 3221, 3519, 4248, 5028, 5542, 6574, 7013, 8093, 8945, 10110, 11043, 12413, 13223, 14476, 15923, 17430, 18617, 20027, 21991, 24016, 25364, 27414, 29356, 31392, 32614, 35743, 37888, 40301, 42620, 45696, 47776, 51109, 53264, 56429, 58471, 61676, 64468, 69437, 72011, 75626

Offset: 1

Scott R. Shannon, Mar 01 2025

nonn

Cf. A381597, A381598, A370708, A281511, A229037.

A381659 Index where n first appears in A381658.

Original entry on oeis.org

1, 4, 11, 21, 31, 69, 93, 96, 174, 222, 263, 433, 529, 633, 671, 732, 1006, 1298, 1388, 1519, 1688, 1813, 2018, 2220, 2507, 2788, 3601, 3949, 4155, 4498, 4612, 4792, 5018, 5476, 5864, 6165, 6391, 6611, 8402, 9173, 9527, 10388, 10727, 11379, 11834, 12045, 12684, 13116, 13552, 14038, 14974, 15340, 15988, 16301, 16994, 18426, 19514, 20372, 21366, 22953

Offset: 1

Scott R. Shannon, Mar 04 2025

nonn

Cf. A381658, A381660, A092482, A381597, A229037.

A236697 First differences of A131741.

Original entry on oeis.org

1, 2, 6, 2, 16, 2, 6, 4, 26, 6, 10, 6, 12, 6, 20, 12, 18, 22, 14, 34, 6, 30, 8, 10, 26, 24, 6, 42, 10, 8, 4, 8, 22, 2, 34, 24, 8, 10, 54, 8, 42, 28, 6, 96, 26, 40, 14, 60, 4, 20, 30, 46, 26, 12, 42, 28, 2, 70, 8, 126, 4, 26, 34, 6, 42, 18, 96, 26, 48, 4

Offset: 1

Zak Seidov, Jan 30 2014

nonn

Among first 10000 terms, the largest is a(7790) = 17412.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A131741, A140577, A185256, A208746, A229037.

a(n) = A131741(n+1) - A131741(n).

A262943 Inverse of conjectured permutation A262942.

Original entry on oeis.org

1, 2, 6, 3, 4, 8, 7, 5, 12, 9, 10, 14, 15, 11, 17, 13, 22, 18, 16, 19, 20, 23, 29, 24, 25, 21, 26, 28, 31, 38, 27, 30, 44, 32, 37, 43, 33, 34, 48, 35, 36, 39, 46, 54, 45, 40, 41, 49, 47, 50, 57, 52, 53, 42, 51, 59, 56, 63, 55, 58, 60, 61, 75, 65, 73, 67

Offset: 1

Max Barrentine, Oct 05 2015

nonn

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A262942, A229037.

N:= 1000: # to get all terms of A262942 before the first > N
V:= Vector(N):
W:= Vector(N):
S:= Vector(N):
firstav:= 1;
for n from 1 to N do
    forbid:= {seq(op([2*V[k]-V[2*k-n], 2*V[2*k-n]-V[k], (V[k]+V[2*k-n])/2]), k=ceil((n+1)/2)..n-1)};
    for v from firstav to N do
      if S[v] <> 0 and v = firstav then firstav:= v+1 fi;
      if S[v] = 0 and not member(v, forbid) then
        V[n]:= v;
        W[v]:= n;
        S[v]:= 1;
        break
      fi
    od;
    if v > N then break fi;
od:
seq(W[i],i=1..firstav-1); # Robert Israel, Nov 23 2015

Corrected and extended by Robert Israel, Nov 23 2015

A306717 Square array T(n, k) of positive integers, n > 0, k > 0, read by antidiagonals, filled the greedy way, such that for any i >= 0 and j >= 0 with i + j > 0, no three terms T(n, k), T(n+i, k+j), T(n+2i, k+2j) form an arithmetic progression.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 4, 2, 3, 1, 1, 3, 2, 4, 4, 4, 5, 2, 1, 2, 5, 4, 4, 1, 4, 5, 4, 2, 2, 4, 5, 4, 1, 1, 1, 7, 2, 4, 3, 4, 2, 7, 1, 1, 2, 1, 2, 5, 4, 5, 5, 4, 5, 2, 1, 2, 1, 2, 5, 1, 5, 5, 4, 5, 5

Offset: 1

Rémy Sigrist, Mar 06 2019

This sequence is a 2-dimensional variant of A229037.

Rémy Sigrist, Colored representation of T(n, k) for n = 1..1000 and k = 1..1000 (where the hue is function of T(n, k))
Rémy Sigrist, C++ program for A306717

Cf. A229037.

T(n, k) = T(k, n).

T(n, 1) = T(n, 2) = A229037(n).

A348190 Positive integers where each is chosen to be the second smallest number subject to the condition that no three terms a(j), a(j+k), a(j+2*k) (for any j and k) form an arithmetic progression.

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 4, 2, 2, 5, 3, 4, 3, 5, 5, 7, 5, 2, 4, 2, 2, 5, 4, 6, 3, 2, 9, 5, 9, 3, 6, 10, 9, 9, 6, 5, 7, 4, 12, 11, 11, 2, 6, 4, 8, 3, 4, 6, 7, 13, 11, 5, 5, 6, 4, 8, 10, 9, 13, 4, 13, 4, 6, 6, 2, 11, 5, 4, 6, 11, 18, 9, 15, 2, 15, 12

Offset: 1

Albert Böschow, Oct 06 2021

The sequence seems to behave in a similar way as the "forest fire" A229037. The graph (up to n=5000) looks like it has a fractal structure, with each dense "pillar" approximately double the size of the previous one.

The terms of this sequence do not seem to be larger (on average) than those of A229037, despite the construction of this sequence.

			a(7) = 4, because 2 would form an arithmetic progression with a(1) = 2 and a(4) = 2 and 3 would form an arithmetic progression with a(5) = 3 and a(6) = 3. Therefore, 4 is the second smallest number which satisfies the condition (1 being the smallest).

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..8000
Rémy Sigrist, PARI program for A348190
Neal Gersh Tolunsky, Scatterplot for n=1...8000

Cf. A229037.

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A359652 Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic or geometric progression.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 5, 5, 1, 1, 2, 1, 1, 2, 2, 5, 5, 2, 5, 5, 6, 6, 10, 6, 6, 11, 1, 1, 2, 1, 1, 2, 2, 5, 5, 1, 1, 2, 1, 1, 2, 2, 5, 5, 2, 5, 5, 6, 6, 10, 6, 6, 11, 11, 5, 5, 6, 6, 12, 6, 6, 12, 2, 12, 14, 13, 13, 3, 13, 12, 12, 15, 13, 12, 12, 13, 15, 13, 17

Offset: 1

Neal Gersh Tolunsky, Jan 09 2023

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

Cf. A229037, A268811.

a(n) <= (n^2+1)/2.

More terms from Alois P. Heinz, Jan 09 2023

A363554 a(1) = 1; for n > 1, a(n) is the smallest positive integer such that both the gradients and y-intercepts of the lines between any two points (i, a(i)) and (j, a(j)) are distinct.

Original entry on oeis.org

1, 1, 2, 5, 11, 4, 3, 18, 26, 35, 48, 66, 16, 99, 129, 27, 67, 149, 190, 8, 235, 259, 285, 348, 276, 34, 24, 97, 362, 170, 155, 15, 504, 464, 9, 639, 449, 173, 391, 768, 577, 682, 836, 937, 598, 438, 94, 6, 1063, 1007, 500, 210, 1146, 1303, 1390, 806, 1530, 62, 1096, 1739, 212, 28, 1001, 1380

Offset: 1

Scott R. Shannon, Jun 10 2023

nonn

This is a variation of A286091 where the y-intercepts of all lines are also distinct.

			a(12) = 66. A value of 15, with coordinate (12,15), for this term would create a point for which all line gradients are distinct, see A286091, but it creates a line that passes through the origin with a(4), a point with coordinate (4,5). However the terms a(3), at coordinate (3,2) and a(6), at coordinate (6,4), have already created a line that passes through the origin, thus a(12) cannot be 15. The coordinate (12,66) is the first point the leads to all lines and y-intercepts being distinct.

Scott R. Shannon, Table of n, a(n) for n = 1..600

Cf. A286091, A236335, A229037.

cp's OEIS Frontend

A381598 Index of first term of three consecutive n's in A381597.

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A381599 Index where n first appears in A381597.

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A381659 Index where n first appears in A381658.

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A236697 First differences of A131741.

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A262943 Inverse of conjectured permutation A262942.

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A306717 Square array T(n, k) of positive integers, n > 0, k > 0, read by antidiagonals, filled the greedy way, such that for any i >= 0 and j >= 0 with i + j > 0, no three terms T(n, k), T(n+i, k+j), T(n+2*i, k+2*j) form an arithmetic progression.

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A348190 Positive integers where each is chosen to be the second smallest number subject to the condition that no three terms a(j), a(j+k), a(j+2*k) (for any j and k) form an arithmetic progression.

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PARI

A359652 Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic or geometric progression.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A363554 a(1) = 1; for n > 1, a(n) is the smallest positive integer such that both the gradients and y-intercepts of the lines between any two points (i, a(i)) and (j, a(j)) are distinct.

Views

Author

Keywords

Comments

Examples

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Crossrefs

A306717 Square array T(n, k) of positive integers, n > 0, k > 0, read by antidiagonals, filled the greedy way, such that for any i >= 0 and j >= 0 with i + j > 0, no three terms T(n, k), T(n+i, k+j), T(n+2i, k+2j) form an arithmetic progression.