A066720 -id:A066720 - cp's OEIS frontend

A066657 Numerators of rational numbers produced in order by A066720(j)/A066720(i) for i >= 1, 1 <= j

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 5, 1, 1, 3, 5, 7, 1, 2, 3, 5, 7, 8, 1, 2, 3, 5, 7, 8, 11, 1, 2, 3, 5, 7, 8, 11, 13, 1, 1, 1, 5, 7, 4, 11, 13, 17, 1, 2, 3, 5, 7, 8, 11, 13, 17, 18, 1, 2, 3, 5, 7, 8, 11, 13, 17, 18, 19, 1, 2, 3, 5, 7, 8, 11, 13, 17, 18, 19, 23, 1, 2, 3, 5, 7

Offset: 0

N. J. A. Sloane, Jan 18 2002

Does every rational number in range (0,1) appear?

a(0) = 1 by convention.

			Sequence of rationals begins 1, 1/2, 1/3, 2/3, 1/5, 2/5, 3/5, 1/7, 2/7, 3/7, 5/7, 1/8, 1/4, 3/8, 5/8, 7/8, 1/11, 2/11, ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A066658 (denominators), A066720.

import Data.List (inits)
import Data.Ratio ((%), numerator)
a066657 n = a066657_list !! n
a066657_list = map numerator
   (1 : (concat $ tail $ zipWith (\u vs -> map (% u) vs)
                                 a066720_list (inits a066720_list)))
-- Reinhard Zumkeller, Nov 19 2013

nmax = 14;
b[1] = 1; F = {1};
For[n = 2, n <= nmax, n++,
For[k = b[n-1]+1, True, k++, Fk = Join[{k^2}, Table[b[i]*k, {i, 1, n-1}]] // Union; If[Fk~Intersection~F == {}, b[n] = k; F = F~Union~Fk; Break[]]]];
Join[{1}, Table[b[k]/b[n], {n, 1, nmax}, {k, 1, n-1}]] // Flatten // Numerator (* Jean-François Alcover, Aug 23 2022, after _Robert Israel in A066720 *)

A066658 Denominators of rational numbers produced in order by A066720(j)/A066720(i) for i >= 1, 1 <= j

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 5, 7, 7, 7, 7, 8, 4, 8, 8, 8, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 17, 18, 9, 6, 18, 18, 9, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29

Offset: 0

N. J. A. Sloane, Jan 18 2002

Does every rational number in range (0,1) appear?

a(0) = 1 by convention.

			Sequence of rationals begins 1, 1/2, 1/3, 2/3, 1/5, 2/5, 3/5, 1/7, 2/7, 3/7, 5/7, 1/8, 1/4, 3/8, 5/8, 7/8, 1/11, 2/11, ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A066657 (numerators), A066720.

import Data.List (inits)
import Data.Ratio ((%), denominator)
a066658 n = a066658_list !! n
a066658_list = map denominator
   (1 : (concat $ tail $ zipWith (\u vs -> map (% u) vs)
                                 a066720_list (inits a066720_list)))
-- Reinhard Zumkeller, Nov 19 2013

nmax = 14;
b[1] = 1; F = {1};
For[n = 2, n <= nmax, n++,
For[k = b[n - 1] + 1, True, k++, Fk = Join[{k^2}, Table[b[i]*k, {i, 1, n - 1}]] // Union; If[Fk~Intersection~F == {}, b[n] = k; F = F~Union~Fk; Break[]]]];
Join[{1}, Table[b[k]/b[n], {n, 1, nmax}, {k, 1, n - 1}]] // Flatten // Denominator (* Jean-François Alcover, Aug 23 2022, after Robert Israel in A066720 *)

A066721 Nonprimes in A066720.

Original entry on oeis.org

1, 8, 18, 50, 60, 81, 98, 105, 128, 242, 264, 308, 338, 416, 495, 520, 546, 560, 578, 625, 663, 675, 684, 864, 935, 952, 1029, 1058, 1083, 1224, 1242, 1254, 1425, 1430, 1682, 1729, 1748, 1771, 1827, 1922, 2436, 2691, 2697, 2720, 2738, 2755, 2790, 2975

Offset: 1

N. J. A. Sloane, Jan 15 2002

2*p^2 is in the sequence for all primes except those in A066775.

David Applegate, First 48186 terms of A066721
David Applegate, C program for computing this sequence
David Applegate, First 48186 terms of A066721 and their factorizations

Cf. A066720, A066775.

Cf. A010051.

a066721 n = a066721_list !! (n-1)
a066721_list = filter ((== 0) . a010051') a066720_list
-- Reinhard Zumkeller, Nov 19 2013

s={1}; xok := Module[ {}, For[ i=1, i<=n, i++, For[ j=1; k=Length[ dl=Divisors[ s[[ i ]]x ]], j<=k, j++; k--, If[ MemberQ[ s, dl[[ j ] ]]&&MemberQ[ s, dl[[ k ] ]], Return[ False ]] ]]; True ]; For[ n=1, True, n++, If[ !PrimeQ[ s[[ n ] ]], Print[ s[[ n ]] ]]; For[ x=s[[ n ]]+1, True, x++, If[ xok, AppendTo[ s, x ]; Break[ ]] ]]

{a066721(m) = local(a,rat,n,s,new,b,i,k,j); a=[]; rat=Set([]); n=0; s=0; while(sKlaus Brockhaus, Feb 23 2002

More terms from Dean Hickerson, Klaus Brockhaus and David Applegate, Jan 15 2002

A066848 Consider sequence of fractions A066657/A066658 produced by ratios of terms in A066720; let m = smallest integer such that all fractions 1/n, 2/n, ..., (n-1)/n have appeared when we reach A066720(m) = k; sequence gives values of k; set a(n) = -1 if some fraction i/n never appears.

Original entry on oeis.org

1, 2, 3, 2436, 520, 60, 308, 2436, 15867, 61800, 8096, 55620, 77077, 20216, 51675, 2296992, 21607, 15867, 185820, 481680, 140805, 226644, 145866, 1568928, 1076000, 187772, 5596587, 1831956, 715778, 3540060, 836535, 2296992, 3088008, 1129514, 7096775, 1995048, 2018646, 3159168, 13019136, 15293320, 6936667, 11250624, 6877463, 20475136, 3380040, 3986360, 1052424, 17566608, 5152350, 1076000, 3824694, 8897564, 2987239, 17600004, 24056230, 89537336, 23397531, 2791424, 5393780, 19344660, 5306268, 8679008, 126415359, 30486400, 29303235

Offset: 1

N. J. A. Sloane, Jan 21 2002

			3/4 does not occur until we reach A066720(401) = 2436 and then we see A066720(320)/A066720(401) = 1827/2436 = 3/4. Therefore a(4) = 2436.

Cf. A066720, A066657, A066658. A066849 gives values of m.

Corrected by John W. Layman, Feb 05 2002

Greatly extended by David Applegate, Feb 13 2002

A066849 Consider sequence of fractions A066657/A066658 produced by ratios of terms in A066720; let m = smallest integer such that all fractions 1/n, 2/n, ..., (n-1)/n have appeared when we reach A066720(m) = k; sequence gives values of m; set a(n) = -1 if some fraction i/n never appears.

Original entry on oeis.org

1, 2, 3, 401, 113, 22, 75, 401, 1986, 6547, 1110, 5949, 7952, 2445, 5578, 172617, 2590, 1986, 17471, 41341, 13631, 20900, 14063, 121563, 86009, 17648, 392866, 140171, 59293, 257162, 68370, 172617, 226693, 89942, 489653, 151601, 153287, 231508, 860521, 999664, 479352, 751180, 475540, 1312350, 246470, 287004, 84285, 1137690, 363942, 86009, 276267, 603972, 219888, 1139722, 1525515, 5227193, 1486480, 206546, 379708, 1244626, 374003, 590152, 7230480, 1903679, 1834403

Offset: 1

N. J. A. Sloane, Jan 21 2002

			3/4 does not occur until we reach A066720(401) = 2436 and then we see A066720(320)/A066720(401) = 1827/2436 = 3/4. Therefore a(4) = 401.

Cf. A066720, A066657, A066658. A066848 gives values of k.

Corrected by John W. Layman, Feb 05 2002

Greatly extended by David Applegate, Feb 13 2002

A079851 Duplicate of A066720.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 17, 18, 19, 23, 29, 31, 37, 41, 43, 47, 50, 53, 59, 60, 61, 67, 71, 73, 79, 81, 83, 89, 97, 98, 101, 103, 105, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 242

Offset: 1

dead

A011185 A B_2 sequence: a(n) = least value such that sequence increases and pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 21, 30, 39, 53, 74, 95, 128, 152, 182, 212, 258, 316, 374, 413, 476, 531, 546, 608, 717, 798, 862, 965, 1060, 1161, 1307, 1386, 1435, 1556, 1722, 1834, 1934, 2058, 2261, 2497, 2699, 2874, 3061, 3197, 3332, 3629, 3712, 3868, 4140, 4447, 4640

Offset: 1

Dan Hoey

nonn

a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j)-a(k) for distinct i and j with 1 <= i,j,k <= n-1. [Comment corrected by Jean-Paul Delahaye, Oct 02 2020.]

Klaus Brockhaus and Dan Hoey, Table of n, a(n) for n = 1..2839
Index entries for B_2 sequences.

Cf. A010672.
Cf. A033627, A003278, A026471, A066720, A075122, A075123, A133097.
Cf. A036241, A349777.

from itertools import islice
def agen(): # generator of terms
    aset, sset, k = set(), set(), 0
    while True:
        k += 1
        while any(k+an in sset for an in aset): k += 1
        yield k; sset.update(k+an for an in aset); aset.add(k)
print(list(islice(agen(), 51))) # Michael S. Branicky, Feb 05 2023

a(n) = A010672(n-1)+1.

A067992 a(0)=1 and, for n > 0, a(n) is the smallest positive integer such that the ratios min(a(k)/a(k-1), a(k-1)/a(k)) for 0 < k <= n are all distinct.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 3, 5, 1, 6, 5, 2, 7, 1, 8, 3, 7, 4, 5, 7, 6, 11, 1, 9, 2, 11, 3, 10, 1, 12, 5, 8, 7, 9, 4, 11, 5, 9, 8, 11, 7, 10, 9, 11, 10, 13, 1, 14, 3, 13, 2, 15, 1, 16, 3, 17, 1, 18, 5, 13, 4, 15, 7, 12, 11, 13, 6, 17, 2, 19, 1, 20, 3, 19, 4, 17, 5, 14, 9, 13, 7, 16, 5, 19, 6, 23, 1, 21, 2

Offset: 0

John W. Layman, Feb 06 2002

Every positive rational number appears exactly once as the ratio of adjacent terms (in either order). Conjecture: adjacent terms are always relatively prime. - Franklin T. Adams-Watters, Sep 13 2006

			The sequence of all rational numbers between 0 and 1 obtained by taking ratios of sorted consecutive terms begins: 1/2, 2/3, 1/3, 1/4, 3/4, 3/5, 1/5, 1/6, 5/6, 2/5, 2/7, 1/7, 1/8, 3/8, 3/7, 4/7, 4/5, 5/7, 6/7. - _Gus Wiseman_, Aug 30 2018

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Neil Calkin and Herbert S. Wilf, Recounting the rationals, The American Mathematical Monthly, Vol. 107, No. 4 (2000), 360-363.
Neil Calkin and Herbert S. Wilf, Recounting the rationals, Fermat's Library (2008).

See A066720 for a somewhat similar sequence.

Cf. A002487, A025582, A050376, A057979.

Nest[Function[seq,Append[seq,NestWhile[#+1&,1,MemberQ[Divide@@@Sort/@Partition[seq,2,1],Min[Last[seq]/#,#/Last[seq]]]&]]],{1},100] (* Gus Wiseman, Aug 30 2018 *)

seen = Set([]); other(p) = for (v=1, oo, my (r = min(v,p)/max(v,p)); if (!set search(seen, r), seen = set union(seen, Set([r])); return (v)))
for (n=0, 88, v = if (n==0, 1, other(v)); print1 (v ", ")) \\ Rémy Sigrist, Aug 07 2017

a(6)=3, since 1/4 and 2/4 = 1/2 have already occurred as ratios of adjacent terms.

A337655 a(1)=1; thereafter, a(n) is the smallest number such that both the addition and multiplication tables for (a(1),...,a(n)) contain n*(n+1)/2 different entries (the maximum possible).

Original entry on oeis.org

1, 2, 5, 7, 15, 22, 31, 50, 68, 90, 101, 124, 163, 188, 215, 253, 322, 358, 455, 486, 527, 631, 702, 780, 838, 920, 1030, 1062, 1197, 1289, 1420, 1500, 1689, 1765, 1886, 2114, 2353, 2410, 2570, 2686, 2857, 3063, 3207, 3477, 3616, 3845, 3951, 4150, 4480, 4595, 4746, 5030, 5286, 5698, 5999, 6497, 6624, 6938, 7219, 7661, 7838, 8469, 8665, 9198, 9351, 9667, 9966

Offset: 1

Jean-Paul Delahaye, Sep 30 2020

nonn

If one specifies that not only are there n(n+1)/2 distinct numbers in the addition and multiplication tables, but that all n(n+1) numbers are distinct, then the sequence is A337946 - David A. Corneth, Oct 02 2020

Rémy Sigrist, Table of n, a(n) for n = 1..3000 (a(1)-a(101) from Jean-Paul Delahaye, a(102)-a(1000) from Peter Kagey)
Rémy Sigrist, C++ program for A337655

See A337659 and A337660 (for the addition table), and A337661 and A337662 (for the multiplication table).
Cf. A337656, A337657, A337658.
For similar sequences that focus just on the addition or multiplication tables, see A005282 and A066720.
Cf. also A337946.

terms=67;a[1]=b[1]=1;a1=b1={1};Do[k=a[n-1]+1;While[a2=Union@Join[{2k},Array[a@#+k&,n-1]];b2=Union@Join[{k^2},Array[b@#*k&,n-1]];Intersection[a2,a1]!={}||Intersection[b2,b1]!={},k++];a[n]=b[n]=k;a1=Union[a1,a2];b1=Union[b1,b2],{n,2,terms}];Array[a,terms] (* Giorgos Kalogeropoulos, Nov 15 2021 *)

A066724 a(1) = 1, a(2) = 2; for n > 1, a(n) is the least integer > a(n-1) such that the products a(i)*a(j) for 1 <= i < j <= n are all distinct.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 30, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 154, 157, 163, 167, 169, 173, 179, 180, 181, 191, 193, 197, 199, 211

Offset: 1

Robert E. Sawyer (rs.1(AT)mindspring.com), Jan 18 2002

The first 15 terms are the same as A026477; the first 13 terms are the same as A026416.

Contains all primes. - Ivan Neretin, Mar 02 2016

			a(7) is not 10 because we already have 10 = 2*5. Of course all primes appear. a(14) is not 24 because if it were there would be a repeat among the terms a(i)*a(j) for 1 <= i < j <= 14, namely 3*16 = 2*24.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A000028, A026477, A026416, A050376, A084400.

Cf. A080431, A066720.

f[l_List] := Block[{k = 1, p = Times @@@ Subsets[l, {2}]},While[Intersection[p, l*k] != {}, k++ ];Append[l, k]];Nest[f, {1, 2}, 62] (* Ray Chandler, Feb 12 2007 *)

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A066657 Numerators of rational numbers produced in order by A066720(j)/A066720(i) for i >= 1, 1 <= j

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A066658 Denominators of rational numbers produced in order by A066720(j)/A066720(i) for i >= 1, 1 <= j

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A066721 Nonprimes in A066720.

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A066848 Consider sequence of fractions A066657/A066658 produced by ratios of terms in A066720; let m = smallest integer such that all fractions 1/n, 2/n, ..., (n-1)/n have appeared when we reach A066720(m) = k; sequence gives values of k; set a(n) = -1 if some fraction i/n never appears.

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A066849 Consider sequence of fractions A066657/A066658 produced by ratios of terms in A066720; let m = smallest integer such that all fractions 1/n, 2/n, ..., (n-1)/n have appeared when we reach A066720(m) = k; sequence gives values of m; set a(n) = -1 if some fraction i/n never appears.

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A079851 Duplicate of A066720.

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A011185 A B_2 sequence: a(n) = least value such that sequence increases and pairwise sums of distinct elements are all distinct.

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A067992 a(0)=1 and, for n > 0, a(n) is the smallest positive integer such that the ratios min(a(k)/a(k-1), a(k-1)/a(k)) for 0 < k <= n are all distinct.

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A337655 a(1)=1; thereafter, a(n) is the smallest number such that both the addition and multiplication tables for (a(1),...,a(n)) contain n*(n+1)/2 different entries (the maximum possible).

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