A296302 -id:A296302 - cp's OEIS frontend

A319055 Maximum product of an integer partition of n with relatively prime parts.

1, 1, 2, 3, 6, 6, 12, 18, 24, 36, 54, 72, 108, 162, 216, 324, 486, 648, 972, 1458, 1944, 2916, 4374, 5832, 8748, 13122, 17496, 26244, 39366, 52488, 78732, 118098, 157464, 236196, 354294, 472392, 708588, 1062882, 1417176, 2125764, 3188646, 4251528, 6377292

Offset: 1

Gus Wiseman, Sep 09 2018

nonn

After a(7), this appears to be the same as A319054.

Cf. A000740, A000792, A000793, A000837, A001414, A007916, A100953, A281116, A289508, A289509, A296302, A319054, A319057.

Table[Max[Times@@@Select[IntegerPartitions[n],GCD@@#==1&]],{n,20}]

A318731 Number of relatively prime Lyndon compositions (aperiodic necklaces of positive integers) with sum n.

Original entry on oeis.org

1, 0, 1, 2, 5, 7, 17, 27, 54, 93, 185, 324, 629, 1143, 2175, 4050, 7709, 14469, 27593, 52276, 99839, 190371, 364721, 698508, 1342170, 2580165, 4970952, 9585232, 18512789, 35787985, 69273665, 134211600, 260300799, 505278705, 981706783

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(6) = 7 relatively prime Lyndon compositions are 15, 114, 132, 123, 1113, 1122, 11112.
The a(7) = 17 relatively prime Lyndon compositions:
  16, 25, 34,
  115, 142, 124, 133, 223,
  1114, 1213, 1132, 1123, 1222,
  11113, 11212, 11122,
  111112.

Cf. A000740, A000837, A008965, A059966, A100953, A296302.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LyndonQ[#]&&GCD@@#==1&]],{n,10}]

Moebius transform of A059966. Second Moebius transform of A008965.

A318748 Number of integer compositions of n that have only one part or whose consecutive parts are coprime and the last and first part are also coprime.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 43, 82, 151, 285, 535, 1005, 1883, 3533, 6631, 12460, 23407, 43952, 82538, 154999, 291088, 546674, 1026687, 1928118, 3621017, 6800300, 12771086, 23984329, 45042959, 84591339, 158863807, 298348613, 560303342, 1052258402, 1976157510

Offset: 0

Gus Wiseman, Sep 02 2018

nonn

			The a(5) = 13 compositions with adjacent parts coprime:
  (5)
  (41) (14) (32) (23)
  (311) (131) (113)
  (2111) (1211) (1121) (1112)
  (11111)
Missing from this list are (221), (212), and (122).

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000740, A008965, A059966, A100953, A167606, A296302, A318726, A318727, A318728, A318745, A328609.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And@@CoprimeQ@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={concat([1], vector(n, i, i > 1) + sum(k=1, n, b(n, k, (i, j)->gcd(i, j)==1)))} \\ Andrew Howroyd, Nov 01 2019

a(n) = A328609(n) + 1 for n > 1. - Andrew Howroyd, Nov 01 2019

a(21)-a(35) from Alois P. Heinz, Sep 02 2018

Name corrected by Gus Wiseman, Nov 04 2019

A056503 Number of periodic palindromic structures of length n using a maximum of two different symbols.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 14, 16, 26, 32, 51, 64, 100, 128, 198, 256, 392, 512, 778, 1024, 1552, 2048, 3091, 4096, 6176, 8192, 12324, 16384, 24640, 32768, 49222, 65536, 98432, 131072, 196744, 262144, 393472, 524288, 786698, 1048576, 1573376, 2097152, 3146256, 4194304

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.
A periodic palindrome is just a necklace that is equivalent to its reverse. The number of binary periodic palindromes of length n is given by A164090(n). A binary periodic palindrome can only be equivalent to its complement when there are an equal number of 0's and 1's. - Andrew Howroyd, Sep 29 2017
Number of cyclic compositions (necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic cyclic compositions begins:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (113)    (33)      (115)
                    (112)   (122)    (114)     (133)
                    (1111)  (11111)  (222)     (223)
                                     (1122)    (11113)
                                     (11112)   (11212)
                                     (111111)  (11122)
                                               (1111111)
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A179181.
Cf. A016116, A045674, A056508, A164090, A285012.
Cf. A000740, A000837, A008965, A025065, A059966, A242414, A296302, A317085, A317086, A317087, A318731.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1; c[n_] := c[n] = If[EvenQ[n], 2^(n/2-1) + c[n/2], 2^((n-1)/2)];
a[n_?OddQ] := b[n]/2; a[n_?EvenQ] := (1/2)*(b[n] + c[n/2]);
Array[a, 45] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[q,And[Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And],Array[SameQ[RotateRight[q,#],Reverse[RotateRight[q,#]]]&,Length[q],1,Or]]]]],{n,15}] (* Gus Wiseman, Sep 16 2018 *)

a(2n+1) = A164090(2n+1)/2 = 2^n, a(2n) = (A164090(2n) + A045674(n))/2. - Andrew Howroyd, Sep 29 2017

a(17)-a(45) from Andrew Howroyd, Apr 07 2017

A296772 Triangle read by rows in which row n lists the compositions of n ordered first by decreasing length and then reverse-lexicographically.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 20 2017

The ordering of compositions in each row is consistent with the reverse-Mathematica ordering of expressions (cf. A124734).

Length of k-th composition is A124748(k-1)+1. - Andrey Zabolotskiy, Dec 20 2017

			Triangle of compositions begins:
(1),
(11),(2),
(111),(21),(12),(3),
(1111),(211),(121),(112),(31),(22),(13),(4),
(11111),(2111),(1211),(1121),(1112),(311),(221),(212),(131),(122),(113),(41),(32),(23),(14),(5).

Cf. A066099, A101211, A108730, A124734, A124748, A228369, A281013, A294859, A296302, A296656, A296773, A296774.

Table[Reverse[Sort[Join@@Permutations/@IntegerPartitions[n]]],{n,6}]

A296773 Triangle read by rows in which row n lists the compositions of n ordered first by decreasing length and then lexicographically.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 20 2017

			Triangle of compositions begins:
(1),
(11),(2),
(111),(12),(21),(3),
(1111),(112),(121),(211),(13),(22),(31),(4),
(11111),(1112),(1121),(1211),(2111),(113),(122),(131),(212),(221),(311),(14),(23),(32),(41),(5).

Cf. A066099, A101211, A108730, A124734, A228369, A281013, A294859, A296302, A296656, A296772, A296774.

Table[Sort[Join@@Permutations/@IntegerPartitions[n],Or[Length[#1]>Length[#2],Length[#1]===Length[#2]&&OrderedQ[{#1,#2}]]&],{n,6}]

A299151 Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 2, 7, 8, 14, 32, 121, 126, 248, 512, 1003, 2048, 4064, 8176, 130539, 32768, 65382, 131072, 261868, 524224, 1048064, 2097152, 4193131, 8388576, 16775168, 33554180, 67104688, 134217728, 268426672, 536870912, 8589802359, 2147482624, 4294934528, 8589934336, 17179801257, 34359738368, 68719345664, 137438949376, 274877643724, 549755813888

Offset: 1

Gus Wiseman, Feb 03 2018

Numerators of rational valued sequence f whose Dirichlet convolution with itself yields function g(n) = A000079(n-1) = 2^(n-1). - Antti Karttunen, Aug 10 2018

			Sequence begins: 1, 1, 2, 7/2, 8, 14, 32, 121/2, 126, 248, 512, 1003, 2048, 4064, 8176, 130539/8, 32768.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000740, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299119, A299149, A299152.

Cf. also A317831.

nn=50;
sys=Table[2^(n-1)==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]

A299151perA299152(n) = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299151perA299152(d)*A299151perA299152(n/d),0)))/2);
A299151(n) = numerator(A299151perA299152(n));

More terms from Antti Karttunen, Jul 29 2018

A318727 Number of integer compositions of n where adjacent parts are indivisible (either way) and the last and first part are also indivisible (either way).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 3, 5, 13, 9, 23, 15, 37, 45, 63, 115, 131, 207, 265, 415, 603, 823, 1251, 1673, 2521, 3519, 5147, 7409, 10449, 15225, 21497, 31285, 44719, 64171, 92315, 131619, 190085, 271871, 391189, 560979, 804265, 1155977, 1656429, 2381307, 3414847

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(10) = 13 compositions:
  (10)
  (7,3) (3,7) (6,4) (4,6)
  (5,3,2) (5,2,3) (3,5,2) (3,2,5) (2,5,3) (2,3,5)
  (3,2,3,2) (2,3,2,3)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000740, A008965, A167606, A285573, A296302, A303362, A304713, A316476, A318726.

Table[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,({_,x_,y_,_}/;Divisible[x,y]||Divisible[y,x])|({y_,_,x_}/;Divisible[x,y]||Divisible[y,x])]&]//Length,{n,20}]

b(n,k,pred)={my(M=matrix(n,n)); for(n=1, n, M[n,n]=pred(k,n); for(j=1, n-1, M[n,j]=sum(i=1, n-j, if(pred(i,j), M[n-j,i], 0)))); sum(i=1, n, if(pred(i,k), M[n,i], 0))}
a(n)={1 + sum(k=1, n-1, b(n-k, k, (i,j)->i%j<>0&&j%i<>0))} \\ Andrew Howroyd, Sep 08 2018

a(21)-a(28) from Robert Price, Sep 07 2018

Terms a(29) and beyond from Andrew Howroyd, Sep 08 2018

A318745 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being coprime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 19, 32, 53, 94, 158, 279, 480, 847, 1487, 2647, 4676, 8349, 14865, 26630, 47700, 85778, 154290, 278318, 502437, 908880, 1645713, 2984546, 5417743, 9847189, 17914494, 32625523, 59467893, 108493134, 198089610, 361965238, 661883231, 1211161991

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(7) = 12 Lyndon compositions with adjacent parts coprime:
  (7)
  (16) (25) (34)
  (115)
  (1114) (1213) (1132) (1123)
  (11113) (11212)
  (111112)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A100953, A167606, A296302, A318728, A318731, A318746, A318747.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,LyndonQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->gcd(i, j)==1))); vector(n, n, (n > 1) + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

a(n) = A328669(n) + 1 for n > 1. - Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A296656 Triangle whose n-th row is the concatenated sequence of all Lyndon compositions of n in reverse-lexicographic order.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 18 2017

			Triangle of Lyndon compositions begins:
(1),
(2),
(3),(12),
(4),(13),(112),
(5),(23),(14),(122),(113),(1112),
(6),(24),(15),(132),(123),(114),(1122),(1113),(11112),
(7),(34),(25),(223),(16),(142),(133),(124),(1222),(1213),(115),(1132),(1123),(11212),(1114),(11122),(11113),(111112).

Cf. A000740, A001037, A001045, A008965, A059966, A060223, A066099, A101211, A102659, A124734, A185700, A228369, A281013, A294859, A296302, A296373.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n],LyndonQ],OrderedQ[PadRight[{#2,#1}]]&],{n,7}]

Row n is a concatenation of A059966(n) Lyndon words with total length A000740(n).

cp's OEIS Frontend

A319055 Maximum product of an integer partition of n with relatively prime parts.

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A318731 Number of relatively prime Lyndon compositions (aperiodic necklaces of positive integers) with sum n.

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A318748 Number of integer compositions of n that have only one part or whose consecutive parts are coprime and the last and first part are also coprime.

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A056503 Number of periodic palindromic structures of length n using a maximum of two different symbols.

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A296772 Triangle read by rows in which row n lists the compositions of n ordered first by decreasing length and then reverse-lexicographically.

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A296773 Triangle read by rows in which row n lists the compositions of n ordered first by decreasing length and then lexicographically.

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Mathematica

A299151 Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

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A318727 Number of integer compositions of n where adjacent parts are indivisible (either way) and the last and first part are also indivisible (either way).

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