A318726
Number of integer compositions of n that have only one part or whose consecutive parts are indivisible and the last and first part are also indivisible.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 5, 3, 8, 13, 12, 23, 27, 56, 64, 100, 150, 216, 325, 459, 700, 1007, 1493, 2186, 3203, 4735, 6929, 10243, 14952, 22024, 32366, 47558, 69906, 102634, 150984, 221713, 325919, 478842, 703648, 1034104, 1519432, 2233062, 3281004, 4821791, 7085359
Offset: 1
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)
The a(11) = 12 compositions:
(11)
(9,2) (2,9) (8,3) (3,8) (7,4) (4,7) (6,5) (5,6)
(5,2,4) (4,5,2) (2,4,5)
-
Table[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,({_,x_,y_,_}/;Divisible[x,y])|({y_,_,x_}/;Divisible[x,y])]&]//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))} \\ Andrew Howroyd, Sep 08 2018
A318728
Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose adjacent parts (including the last with first) are coprime.
Original entry on oeis.org
1, 2, 3, 4, 6, 9, 13, 22, 34, 58, 95, 168, 280, 492, 853, 1508, 2648, 4715, 8350, 14924, 26643, 47794, 85779, 154475, 278323, 502716, 908913, 1646206, 2984547, 5418653, 9847190, 17916001, 32625618, 59470540, 108493150, 198094483, 361965239, 661891580, 1211162271
Offset: 1
The a(7) = 13 cyclic compositions with adjacent parts coprime:
7,
16, 25, 34,
115,
1114, 1213, 1132, 1123,
11113, 11212,
111112,
1111111.
-
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&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, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019
A318729
Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose consecutive parts (including the last with first) are indivisible.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 2, 4, 6, 6, 8, 11, 19, 21, 30, 41, 59, 79, 112, 157, 219, 305, 430, 605, 860, 1210, 1727, 2424, 3463, 4905, 7001, 9954, 14211, 20271, 28980, 41392, 59254, 84800, 121540, 174163, 249932, 358578, 515091, 739933, 1063827, 1529767, 2201383
Offset: 1
The a(13) = 11 cyclic compositions with successive parts indivisible:
(13)
(2,11) (3,10) (4,9) (5,8) (6,7)
(2,4,7) (2,6,5) (2,8,3) (3,6,4)
(2,3,5,3)
Cf.
A000740,
A008965,
A059966,
A167606,
A285573,
A303362,
A304713,
A316476,
A318726,
A318727,
A318728,
A318730,
A328600.
-
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&And@@Not/@Divisible@@@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)->i%j<>0))); vector(n, n, 1 + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019
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
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).
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
A318730
Number of cyclic compositions (necklaces of positive integers) summing to n with adjacent parts (including the last and first part) being indivisible (either way).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 2, 3, 6, 5, 8, 7, 14, 15, 21, 31, 39, 51, 69, 98, 133, 177, 254, 329, 471, 632, 902, 1230, 1710, 2370, 3270, 4591, 6384, 8898, 12429, 17252, 24230, 33783, 47405, 66254, 92860, 130142, 182469, 256262, 359676, 505231, 710059, 997953, 1404215
Offset: 1
The a(14) = 14 cyclic compositions with adjacent parts indivisible either way:
(14)
(3,11) (4,10) (5,9) (6,8)
(2,5,7) (2,7,5) (3,4,7) (3,7,4)
(2,3,2,7) (2,3,4,5) (2,5,2,5) (2,5,4,3) (3,4,3,4)
Cf.
A000740,
A008965,
A059966,
A167606,
A285573,
A303362,
A304713,
A316476,
A318726,
A318727,
A318728,
A318729,
A328601.
-
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And[neckQ[#],And@@Not/@Divisible@@@Partition[#,2,1,1],And@@Not/@Divisible@@@Reverse/@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)->i%j<>0 && j%i<>0))); vector(n, n, 1 + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019
A318746
Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and successive parts (including the last with the first part) being indivisible.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 2, 4, 5, 6, 8, 11, 17, 20, 29, 41, 56, 79, 107, 155, 214, 305, 422, 604, 850, 1207, 1709, 2424, 3439, 4905, 6972, 9949, 14171, 20268, 28915, 41392, 59176, 84790, 121428, 174163, 249760, 358578, 514873, 739910, 1063523, 1529767, 2200926
Offset: 1
The a(14) = 17 Lyndon compositions with successive parts indivisible:
(14)
(3,11) (4,10) (5,9) (6,8)
(2,3,9) (2,5,7) (2,7,5) (3,4,7) (3,6,5) (3,7,4)
(2,3,2,7) (2,3,4,5) (2,4,3,5) (2,4,5,3) (2,5,4,3)
(2,3,2,4,3)
Cf.
A000740,
A008965,
A059966,
A285573,
A303362,
A318726,
A318727,
A318729,
A318730,
A318731,
A318745,
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@@Not/@Divisible@@@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)->i%j<>0))); vector(n, n, 1 + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019
A318747
Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being indivisible (either way).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 2, 3, 5, 5, 8, 7, 12, 14, 20, 31, 37, 51, 64, 96, 129, 177, 246, 328, 465, 630, 889, 1230, 1692, 2370, 3250, 4587, 6354, 8895, 12384, 17252, 24180, 33777, 47336, 66254, 92752, 130142, 182337, 256246, 359500, 505231, 709787, 997951, 1403883
Offset: 1
The a(14) = 12 Lyndon compositions with adjacent parts indivisible either way:
(14)
(3,11) (4,10) (5,9) (6,8)
(2,5,7) (2,7,5) (3,4,7) (3,7,4)
(2,3,2,7) (2,3,4,5) (2,5,4,3)
Cf.
A000740,
A008965,
A059966,
A285573,
A303362,
A318726,
A318727,
A318729,
A318730,
A318731,
A318745,
A318746.
-
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,And[LyndonQ[#],And@@Not/@Divisible@@@Partition[#,2,1,1],And@@Not/@Divisible@@@Reverse/@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)->i%j<>0 && j%i<>0))); vector(n, n, 1 + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019
Showing 1-7 of 7 results.