A281579 -id:A281579 - cp's OEIS frontend

A281900 Index of first occurrence of n in A281579, or -1 if n does not appear there..

1, 3, 6, 7, 15, 16, 34, 240, 241, 317, 502, 503, 504, 505, 1077, 1178, 1179, 1806, 4866, 4901, 4902, 4946, 4947, 8195, 8196, 8197, 8198, 8199, 8200, 8201, 8202, 8203, 8204, 45393, 45485, 45486, 96358, 96359, 96360, 96361, 96362, 96363, 96364, 96365, 96366

Offset: 2

Rémy Sigrist, Feb 01 2017

nonn

The sequence is strictly increasing.
It is conjectured that every number >= 2 does appear in A281579.
a(n) is also the index of the first occurrence of n-1 in A380317, or -1 if n-1 does not appear there.

Rémy Sigrist, Table of n, a(n) for n = 2..1904 (a(2)-a(968) added on Feb 01 2017, a(969)-a(1904) added on Feb 06 2017)
Rémy Sigrist, C++ program for A281900

Cf. A281579, A310387.

Escape clause added to definition by N. J. A. Sloane, Feb 17 2025

A281772 a(n)=common difference of the n-th run of consecutive terms in arithmetic progression in A281579.

Original entry on oeis.org

0, 1, 0, 1, -1, 0, 1, -1, 0, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, -2, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, 0, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -4, 0, 1, -1, 0, 1, 0, 1, -2, 0, 1, -1, 0, 1

Offset: 1

Rémy Sigrist, Jan 29 2017

sign

All terms are less than or equal to 1, and a(n)=1 implies a(n-1)=0.

There are no consecutive equal terms.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A281579

A281783 a(n)=index of first term of n-th run of consecutive terms in arithmetic progression in A281579.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 16, 19, 21, 23, 25, 27, 30, 34, 39, 45, 50, 54, 57, 60, 63, 65, 66, 67, 68, 70, 73, 76, 79, 82, 86, 91, 97, 104, 110, 115, 119, 122, 124, 126, 128, 130, 132, 134, 136, 139, 143, 148, 154, 161, 167, 172, 176, 179, 182, 185, 188, 192, 197

Offset: 1

Rémy Sigrist, Jan 29 2017

nonn

This sequence is strictly increasing.

For any n>0, a(n+1)=index of last term of n-th run of consecutive terms in arithmetic progression in A281579.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A281579.

a(n)=2-n+Sum_{k=1..n-1} A281579(k) for any n>0.

A362816 Lexicographically earliest sequence such that nowhere is a term a(n) contained in an arithmetic progression of length greater than a(n).

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 3, 5, 2, 2, 3, 2, 2, 3, 3, 3, 5, 3, 5, 5, 5, 3, 3, 3, 5, 5, 2, 2, 3, 2, 2, 5, 5, 3, 3, 2, 2, 3, 2, 2, 5, 3, 3, 5, 3, 5, 5, 3, 3, 5, 5, 3, 5, 5, 5, 6, 5, 3, 5, 5, 6, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 3, 3, 5, 5, 5, 6, 5, 5, 3, 2, 2, 5, 2, 2, 6

Offset: 1

Samuel Harkness, May 04 2023

nonn

Progressions are terms at indices in arithmetic progression and with values which are some arithmetic progression too.
1 is never in the sequence, because if a(n) = 1, then {a(n),a(n+1)} would form an arithmetic progression greater than 1 in length.
Conjecture: only terms in A362815 appear in this sequence. This is true through the first 10^5 terms.
If this is true, then a(A003278) = 2, because the only way to constrain 2 would be {2,2,2}, and A003278 is defined by adding the smallest term which avoids any 3 term arithmetic progressions. If the conjecture is false, arithmetic progressions {4,3,2}, {8,5,2}, etc. may further constrain 2s.

			For n=9 first we check 1 (never in the sequence). If a(9) were 2, {a(1),a(5),a(9)} = {2,2,2} would form an arithmetic progression of length 3 with a minimum value of 2; this is not allowed. Next, if a(9) were 3, {a(6),a(7),a(8),a(9)} = {3,3,3,3} would form an arithmetic progression of length 4 with a minimum value of 3; this is not allowed. Next, if a(9) were 4, {a(5),a(7),a(9)} = {2,3,4} would form an arithmetic progression of length 3 with a minimum value of 2; this is not allowed. Last, a(9) = 5 fits the definition, as no arithmetic progressions p can be made such that length(p) > min (p) and 5 is the least positive integer where this is satisfied, so a(9) = 5.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Samuel Harkness, MATLAB program

Cf. A362815, A363011 (indices of record highs), A003278, A090822, A281579.

MATLAB
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A113138 Self-describing sequence made of strings of consecutive integers. The number of elements in each string is the sequence itself.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Jan 04 2006

The less interesting sequence 1,1,1,1,1,1,1,1, ... obeys the same rule.

			First string is "1", having 1 element. Second string is "3,4,5" having 3 elements. Third string is "1,2,3,4" which has 4 elements. Next string is "1,2,3,4,5", made of 5 elements. Next string is "1", having only 1 element, etc. So we have 1 element, then 3, then 4, then 5, then 1, etc. This is the sequence itself.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, PARI program for A113138

Cf. A281579.

More terms from Rémy Sigrist, Feb 08 2017

A380317 The lexicographically earliest sequence of positive numbers which is identical to the run lengths of its first differences.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 2, 2, 2, 2, 3, 4, 5, 6, 5, 4, 3, 3, 3, 2, 1, 1, 1, 2, 3, 3, 3, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 2, 2, 2, 2, 2, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 3, 3, 3, 4, 5, 6, 5, 4, 3, 3, 3, 1, 1, 2, 2, 2, 3, 4, 5, 4, 3, 2, 2, 2, 2, 3, 4, 5, 4, 3, 2, 1, 1

Offset: 1

Dominic McCarty, Feb 13 2025

34 is the smallest value that does not appear in the first 10000 terms.
Conjecture: Every positive integer eventually appears.
Shortly after submitting this sequence the author, Dominic McCarty, discovered that it is almost identical to A281579: in fact, a(n) = A281579(n) - 1 for all n. However, since A281579 has seniority and the present sequence has a simpler definition and is more likely to be searched for, it has been decided to retain both entries. - N. J. A. Sloane, Feb 17 2025
The index of n in the present sequence is given by A281900(n+1).

			The sequence of first differences (where the n-th term is a(n+1)-a(n)) is:
0, 1, 0, 0, 1, 1, -1, -1, 0, 0, 0, 1, 1, 1, 1, -1, -1, -1, 0, 0, ...
The run lengths of consecutive values are:
1, 1, 2, 2, 2, 3, 4, 3, 2, ...
Which is the original sequence.

Dominic McCarty, Table of n, a(n) for n = 1..10000

Cf. A281579, A281900.

from itertools import groupby
def runs(l): return [len(list(group)) for i, group in groupby(l)]
def firstDifs(l): return [l[i]-l[i-1] for i in range(1,len(l))]
a = [1,1]
while len(runs(firstDifs(a))) <= 100:
    a.append(1)
    b, m = runs(firstDifs(a)), max(firstDifs(a))
    while not (all(b[n] == a[n] for n in range(len(b)-1)) and b[-1] <= a[len(b)-1]):
        a[-1] += 1
        if a[-1] > m+a[-2]+1: a.pop(); a[-1] += 1 #Backtracking needed
        b = runs(firstDifs(a))
print(a[:len(runs(firstDifs(a)))])

A381537 Lexicographically least sequence of natural numbers such that for all arithmetic progressions p, length(p) <= sqrt(max(p)).

Original entry on oeis.org

1, 4, 5, 8, 9, 10, 12, 15, 16, 17, 18, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92

Offset: 1

Samuel Harkness, Feb 26 2025

nonn

Up to a(n) the longest possible arithmetic progression is sqrt(a(n)).

Does the density of this sequence approach 1?

			1 is in the sequence, as 1 creates the arithmetic progression p = {1}, where length(p) = 1 and sqrt(max(p)) = 1.
For 2: the arithmetic progression p = {1,2} would be created. Here, length(p) = 2, and sqrt(max(p)) = sqrt(2), so length(p) > sqrt(max(p)), thus 2 is not in the sequence. Similarly, 3 is not in the sequence.
For 4: p = {1,4} is the only new arithmetic progression. Here, length(p) = 2, and sqrt(max(p)) = 2, so 4 is in the sequence. Similarly, 5 is in the sequence.
For 6: the arithmetic progression p = {4,5,6} would be created. Here, length(p) = 3, and sqrt(max(p)) = sqrt(6), so length(p) > sqrt(max(p)), thus 6 is not in the sequence.

Samuel Harkness, MATLAB program

Cf. A281579, A362815.

MATLAB
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cp's OEIS Frontend

A281900 Index of first occurrence of n in A281579, or -1 if n does not appear there..

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A281772 a(n)=common difference of the n-th run of consecutive terms in arithmetic progression in A281579.

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A281783 a(n)=index of first term of n-th run of consecutive terms in arithmetic progression in A281579.

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A362816 Lexicographically earliest sequence such that nowhere is a term a(n) contained in an arithmetic progression of length greater than a(n).

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A113138 Self-describing sequence made of strings of consecutive integers. The number of elements in each string is the sequence itself.

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A380317 The lexicographically earliest sequence of positive numbers which is identical to the run lengths of its first differences.

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Python

A381537 Lexicographically least sequence of natural numbers such that for all arithmetic progressions p, length(p) <= sqrt(max(p)).

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MATLAB