A065825 -id:A065825 - cp's OEIS frontend

A225745 Smallest k such that n numbers can be picked in {1,...,k} with no four in arithmetic progression.

1, 2, 3, 5, 6, 8, 9, 10, 13, 15, 17, 19, 21, 23, 25, 27, 28, 30, 33, 34, 37, 40, 43, 45, 48, 50, 53, 54, 58, 60, 64, 66, 68, 70, 74, 77, 79, 82, 84, 87, 91

Offset: 1

Don Knuth, Aug 05 2013

			a(8)=10 because of the unique solution 1 2 3 5 6 8 9 10.

Knuth, Donald E., Satisfiability, Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, pages 135 and 190, Problem 31.

Samuel S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771.

This sequence is to A003003 as A065825 is to A003002.

a(37)-a(41) from Bert Dobbelaere, Sep 16 2020

A133234 a(n) is least semiprime (not already in list) such that no 3-term subset forms an arithmetic progression.

Original entry on oeis.org

4, 6, 9, 10, 15, 22, 25, 33, 39, 49, 55, 58, 82, 86, 87, 93, 111, 118, 121, 122, 134, 145, 185, 194, 201, 202, 206, 215, 237, 247, 274, 287, 298, 299, 303, 305, 314, 334, 335, 358, 362, 386, 446, 447, 454, 471, 482, 497, 502, 527, 529, 537, 553, 554, 562, 614

Offset: 1

Jonathan Vos Post, Oct 13 2007

This is to semiprimes A001358 as A131741 is to primes A000040.

Cf. A000040, A001358, A065825, A131741.

NextSemiprime[n_] := Block[{c = n + 1, f = 0}, While[Plus @@ Last /@ FactorInteger[c] != 2, c++ ]; c ]; f[l_List] := Block[{c, f = 0}, c = If[l == {}, 2, l[[ -1]]]; While[f == 0, c = NextSemiprime[c]; If[Intersection[l, l - (c - l)] == {}, f = 1]; ]; Append[l, c] ]; Nest[f, {}, 100] (* Ray Chandler, Nov 10 2007 *)

a(1) = 4, a(2) = 6, a(n) = smallest semiprime such that there is no i < j < n with a(n) - a(j) = a(j) - a(i).

More terms from Ray Chandler, Nov 10 2007

A225859 Smallest k such that n numbers can be picked in {1,...,k} with no five terms in arithmetic progression.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 24, 25, 27, 28, 29, 31, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 49, 51, 52, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 76

Offset: 1

Don Knuth, Aug 05 2013

Knuth, Donald E., Satisfiability, Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, pages 135 and 190, Problem 31.

Samuel S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771.

This sequence is to A003004 as A065825 is to A003002.

Cf. A226066.

A226066 Smallest k such that n numbers can be picked in {1,...,k} with no six terms in arithmetic progression.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 29, 32, 33, 35, 36, 37, 39, 40, 41, 44, 45, 46, 48, 49, 50, 51, 54, 56, 58, 59, 61, 62

Offset: 1

Don Knuth, Aug 05 2013

Knuth, Donald E., Satisfiability, Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, pages 135 and 190, Problem 31.

Samuel S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771.

This sequence is to A003005 as A065825 is to A003002.

Cf. A225745, A225859.

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).

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

Joseph Wheat, Dec 21 2019

nonn

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.

Encyclopedia of Mathematics, Density of a sequence
Eric Weisstein's World of Mathematics, Arithmetic Progression

Cf. A049988, A130518, A104429, A065825, A092482.

Partial sums of A002541.

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

a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i - 1)/(j + 1)).

cp's OEIS Frontend

A225745 Smallest k such that n numbers can be picked in {1,...,k} with no four in arithmetic progression.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A133234 a(n) is least semiprime (not already in list) such that no 3-term subset forms an arithmetic progression.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A225859 Smallest k such that n numbers can be picked in {1,...,k} with no five terms in arithmetic progression.

Views

Author

Keywords

References

Links

Crossrefs

A226066 Smallest k such that n numbers can be picked in {1,...,k} with no six terms in arithmetic progression.

Views

Author

Keywords

References

Links

Crossrefs

A236697 First differences of A131741.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula