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.

Showing 1-4 of 4 results.

A328593 Numbers whose binary indices have no consecutive divisible parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 14, 16, 18, 20, 22, 24, 28, 30, 32, 40, 44, 46, 48, 50, 52, 54, 56, 60, 62, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 104, 108, 110, 112, 114, 116, 118, 120, 124, 126, 128, 132, 134, 144, 146, 148, 150, 152, 156, 158, 160
Offset: 1

Views

Author

Gus Wiseman, Oct 21 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Examples

			The sequence of terms together with their binary expansions and binary indices begins:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   4:    100 ~ {3}
   6:    110 ~ {2,3}
   8:   1000 ~ {4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  16:  10000 ~ {5}
  18:  10010 ~ {2,5}
  20:  10100 ~ {3,5}
  22:  10110 ~ {2,3,5}
  24:  11000 ~ {4,5}
  28:  11100 ~ {3,4,5}
  30:  11110 ~ {2,3,4,5}
  32: 100000 ~ {6}
  40: 101000 ~ {4,6}
  44: 101100 ~ {3,4,6}
  46: 101110 ~ {2,3,4,6}
  48: 110000 ~ {5,6}
  50: 110010 ~ {2,5,6}

Crossrefs

The version for prime indices is A328603.
Numbers with no successive binary indices are A003714.
Partitions with no consecutive divisible parts are A328171.
Compositions without consecutive divisible parts are A328460.
Cf. A000120, A014081, A328187, A328508, A328594, A328595, A328598, A328600, A328608.

Programs

  • Mathematica
    Select[Range[0,100],!MatchQ[Join@@Position[Reverse[IntegerDigits[#,2]],1],{_,x_,y_,_}/;Divisible[y,x]]&]

A328600 Number of necklace compositions of n with no part circularly followed by a divisor.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 3, 5, 5, 7, 10, 18, 20, 29, 40, 58, 78, 111, 156, 218, 304, 429, 604, 859, 1209, 1726, 2423, 3462, 4904, 7000, 9953, 14210, 20270, 28979, 41391, 59253, 84799, 121539, 174162, 249931, 358577, 515090, 739932, 1063826, 1529766, 2201382, 3168565
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2019

Keywords

Comments

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.
Circularity means the last part is followed by the first.

Examples

			The a(5) = 1 through a(13) = 18 necklace compositions (empty column not shown):
  (2,3)  (2,5)  (3,5)  (2,7)    (3,7)      (2,9)    (5,7)      (4,9)
         (3,4)         (4,5)    (4,6)      (3,8)    (2,3,7)    (5,8)
                       (2,4,3)  (2,3,5)    (4,7)    (2,7,3)    (6,7)
                                (2,5,3)    (5,6)    (3,4,5)    (2,11)
                                (2,3,2,3)  (2,4,5)  (3,5,4)    (3,10)
                                                    (2,3,2,5)  (2,4,7)
                                                    (2,3,4,3)  (2,6,5)
                                                               (2,8,3)
                                                               (3,6,4)
                                                               (2,3,5,3)

Links

Crossrefs

The non-necklace version is A328598.
The version with singletons is A318729.
The case forbidding multiples as well as divisors is A328601.
The non-necklace, non-circular version is A328460.
The version for co-primality (instead of divisibility) is A328602.
Necklace compositions are A008965.
Partitions with no part followed by a divisor are A328171.
Cf. A032153, A167606, A318748, A328508, A328593, A328599, A328603, A328608, A328609.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]&]],{n,10}]
  • PARI
    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, sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 26 2019

Formula

a(n) = A318729(n) - 1.

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

A328601 Number of necklace compositions of n with no part circularly followed by a divisor or a multiple.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 2, 5, 4, 7, 6, 13, 14, 20, 30, 38, 50, 68, 97, 132, 176, 253, 328, 470, 631, 901, 1229, 1709, 2369, 3269, 4590, 6383, 8897, 12428, 17251, 24229, 33782, 47404, 66253, 92859, 130141, 182468, 256261, 359675, 505230, 710058, 997952, 1404214
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2019

Keywords

Comments

A necklace composition of n (A008965) is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.
Circularity means the last part is followed by the first.

Examples

			The a(5) = 1 through a(13) = 6 necklace compositions (empty column not shown):
  (2,3)  (2,5)  (3,5)  (2,7)  (3,7)      (2,9)  (5,7)      (4,9)
         (3,4)         (4,5)  (4,6)      (3,8)  (2,3,7)    (5,8)
                              (2,3,5)    (4,7)  (2,7,3)    (6,7)
                              (2,5,3)    (5,6)  (3,4,5)    (2,11)
                              (2,3,2,3)         (3,5,4)    (3,10)
                                                (2,3,2,5)  (2,3,5,3)
                                                (2,3,4,3)

Links

Crossrefs

The non-necklace version is A328599.
The case forbidding divisors only is A328600 or A318729 (with singletons).
The non-necklace, non-circular version is A328508.
The version for co-primality (instead of indivisibility) is A328597.
Cf. A000740, A008965, A032153, A167606, A318748, A328171, A328460, A328593, A328598, A328602, A328603, A328608, A328609.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]&&And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]&]],{n,10}]
  • PARI
    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, sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 26 2019

Formula

a(n) = A318730(n) - 1.

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

A328599 Number of compositions of n with no part circularly followed by a divisor or a multiple.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 0, 4, 2, 4, 12, 8, 22, 14, 36, 44, 62, 114, 130, 206, 264, 414, 602, 822, 1250, 1672, 2520, 3518, 5146, 7408, 10448, 15224, 21496, 31284, 44718, 64170, 92314, 131618, 190084, 271870, 391188, 560978, 804264, 1155976, 1656428, 2381306, 3414846
Offset: 0

Views

Author

Gus Wiseman, Oct 25 2019

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
Circularity means the last part is followed by the first.

Examples

			The a(0) = 1 through a(12) = 22 compositions (empty columns not shown):
  ()  (2,3)  (2,5)  (3,5)  (2,7)  (3,7)      (2,9)  (5,7)
      (3,2)  (3,4)  (5,3)  (4,5)  (4,6)      (3,8)  (7,5)
             (4,3)         (5,4)  (6,4)      (4,7)  (2,3,7)
             (5,2)         (7,2)  (7,3)      (5,6)  (2,7,3)
                                  (2,3,5)    (6,5)  (3,2,7)
                                  (2,5,3)    (7,4)  (3,4,5)
                                  (3,2,5)    (8,3)  (3,5,4)
                                  (3,5,2)    (9,2)  (3,7,2)
                                  (5,2,3)           (4,3,5)
                                  (5,3,2)           (4,5,3)
                                  (2,3,2,3)         (5,3,4)
                                  (3,2,3,2)         (5,4,3)
                                                    (7,2,3)
                                                    (7,3,2)
                                                    (2,3,2,5)
                                                    (2,3,4,3)
                                                    (2,5,2,3)
                                                    (3,2,3,4)
                                                    (3,2,5,2)
                                                    (3,4,3,2)
                                                    (4,3,2,3)
                                                    (5,2,3,2)

Links

Crossrefs

The necklace version is A328601.
The case forbidding only divisors (not multiples) is A328598.
The non-circular version is A328508.
Partitions with no part followed by a divisor are A328171.
Cf. A000740, A008965, A167606, A318729, A318748, A328460, A328593, A328600, A328603, A328608, A328609, A328674.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Not/@Divisible@@@Partition[#,2,1,1]&&And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]&]],{n,0,10}]
  • PARI
    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], sum(k=1, n, b(n, k, (i,j)->i%j<>0&&j%i<>0)))} \\ Andrew Howroyd, Oct 26 2019

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019
Showing 1-4 of 4 results.

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