cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A101886 Smallest natural number sequence without any length 4 equidistant arithmetic subsequences.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 14, 15, 16, 18, 19, 20, 22, 24, 27, 28, 29, 31, 32, 35, 36, 37, 39, 41, 42, 43, 47, 48, 50, 51, 53, 55, 58, 60, 61, 63, 65, 66, 68, 70, 71, 72, 77, 78, 80, 82, 85, 86, 87, 89, 90, 91, 94, 95, 96, 98, 99, 100, 102, 103, 104, 107, 109, 110, 111, 114
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 20 2004

Keywords

Examples

			4 is out because of 1,2,3,4. 13 is out because of 1,5,9,13.

Crossrefs

Cf. A101887, A101884, A101888.
A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

A101888 Smallest natural number sequence without any length 5 equidistant arithmetic subsequences.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 82, 86, 87, 88, 90, 91, 92, 93, 95
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 20 2004

Keywords

Examples

			5 is out because of 1,2,3,4,5. 21 is out because of 1,6,11,16,21.

Crossrefs

Cf. A101889, A101884, A101886.
A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

A381658 Lexicographically earliest sequence of positive integers such that for each distinct positive integer t there is only one value of k such that t = a(n) = a(n+k) = a(n+2*k).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 3, 3, 1, 1, 3, 1, 1, 2, 2, 4, 2, 2, 3, 3, 4, 3, 3, 4, 4, 5, 4, 3, 5, 5, 1, 1, 5, 1, 1, 4, 4, 2, 2, 1, 1, 2, 1, 1, 5, 3, 2, 2, 5, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 5, 3, 3, 4, 6, 2, 4, 6, 2, 6, 4, 6, 6, 5, 3, 3, 4, 3, 5, 4, 4, 5, 5, 6, 6, 4, 6, 6, 7, 7, 7, 8, 5, 1, 1, 5, 1, 1, 6, 5, 5, 7, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 8, 4, 6
Offset: 1

Views

Author

Scott R. Shannon, Mar 03 2025

Keywords

Comments

In the first 2.5 million terms the only numbers to appear in three consecutive terms are 1 (at n = 1), 2 (at n = 8), 5 (at n = 11), 7 (at n = 93), 8 (at n = 169), and 112 (at n = 96610). It is unknown if more such numbers exist.
It is conjectured that the values of n for which a(n) = 1 is given by A092482.
See A381660 for the single value of k for each distinct positive integer, and A381659 for the index where each such integer first appears.

Examples

			a(1) = a(2) = a(3) = 1. As 1 has now appeared in three terms satisfying a(n) = a(n+k) = a(n+2*k) = 1, with k = 1 in this instance, no other three terms equalling 1 can appear anywhere in the sequence that would satisfy a similar relationship.
a(4) = a(5) = 2 as choosing 1 would create another three terms equalling 1 separated by 1, and three terms equalling 1 separated by 2, namely a(1), a(3), a(5). As neither of those is permitted, the next smallest number 2 is chosen.
a(6) = 1 as this does not create any three terms equalling 1 separated by any value k, so 1 is again chosen.
a(10) = 2 as choosing 1 would create three terms a(2) = a(6) = a(10) = 1 with a difference of 4 which is not permitted. Note that a(9) = a(10) = a(11) = 2, so no other three terms equalling 2 can appear anywhere in the sequence that would satisfy a(n) = a(n+k) = a(n+2*k) = 2.
a(11) = 3 as choosing 1 would create three terms a(3) = a(7) = a(11) = 1 with a difference of 4, while choosing 2 would create a(9) = a(10) = a(11) = 2 with a difference of 1. As neither is permitted the next smallest number 3 is chosen.

Links

Crossrefs

Cf. A381659 (index of first appearance), A381660 (k values), A092482 (indices of 1's), A381597, A229037.

A381660 The value k such that n = a(j) = a(j+k) = a(j+2*k) in A381658.

Original entry on oeis.org

1, 1, 1, 3, 15, 2, 1, 1, 3, 9, 43, 26, 28, 15, 4, 36, 18, 25, 6, 25, 31, 20, 70, 46, 26, 352, 114, 11, 19, 23, 49, 56, 70, 15, 56, 79, 46, 409, 29, 48, 24, 48, 52, 16, 77, 11, 123, 16, 78, 73, 48, 44, 49, 31, 11, 178, 330, 305, 180, 454, 147, 45, 158, 280, 108, 296, 53, 13, 22, 4, 184, 145, 99, 86, 114, 6, 42, 41, 248, 76, 570, 54, 204, 25, 125, 522, 110
Offset: 1

Views

Author

Scott R. Shannon, Mar 04 2025

Keywords

Comments

The last known term that equals 1 is a(112). See A381658.

Links

Crossrefs

Cf. A381658, A381659, A092482, A381597, 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

Views

Author

Scott R. Shannon, Mar 04 2025

Keywords

Crossrefs

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

Views

Author

Zak Seidov, Jan 30 2014

Keywords

Comments

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

Links

Crossrefs

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

Formula

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

A330285 The maximum number of arithmetic progressions in a sequence of length n.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 20, 29, 41, 55, 72, 90, 113, 137, 164, 194, 228, 263, 303, 344, 390, 439, 491, 544, 604, 666, 731, 799, 872, 946, 1027, 1109, 1196, 1286, 1379, 1475, 1579, 1684, 1792, 1903, 2021, 2140, 2266, 2393, 2525, 2662, 2802, 2943, 3093, 3245, 3402, 3562, 3727
Offset: 1

Views

Author

Joseph Wheat, Dec 21 2019

Keywords

Comments

The partial arithmetic density D_n(A) up to n is merely the number of arithmetic progressions, A(s(n)), divided by the total number of nonempty subsets of {s(1), s(2), ..., s(n)}, i.e., A(s(n))/(2^n - 1). As n approaches infinity, D_n(A) converges to zero. Furthermore, the infinite sum of the partial densities for any sequence always converges to the total density D(A). Every infinite arithmetic progression has the same total density, Sum_{n >= 1} a(n)/(2^n - 1) = alpha ~ 1.25568880818612911696845537; sequences with a finite number of progressions have D(A) < alpha; and sequences without any arithmetic progressions have D(A) = 0.

Links

Crossrefs

Cf. A049988, A130518, A104429, A065825, A092482.
Partial sums of A002541.

Programs

  • PARI
    a(n) = sum(i=1, n, sum(j=1, i, floor((i - 1)/(j + 1))))

Formula

a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i - 1)/(j + 1)).
Previous Showing 11-17 of 17 results.

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