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 1-5 of 5 results.

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 0, 2, 0, 3, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 0, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 0, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 0, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2
Offset: 1

Views

Author

Gus Wiseman, Apr 29 2018

Keywords

Comments

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.
The positions of zeros in this sequence are the prime powers A000961.

Examples

			The a(144) = 8 aperiodic factorizations are (2*2*2*3*6), (2*2*2*18), (2*2*3*12), (2*3*24), (2*6*12), (2*72), (3*48) and (6*24). Missing from this list are (12*12), (2*2*6*6) and (2*2*2*2*3*3).

Crossrefs

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A100953, A275870, A281116, A301700, A303386, A303431, A303546, A303551, A303707, A303709, A303710.

Programs

  • Mathematica
    radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[Select[facsr[n],GCD@@Length/@Split[#]===1&]],{n,100}]

Formula

a(n) = Sum_{d in A007916, d|A052409(n)} mu(d) * A303707(n^(1/d)).

A303974 Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.

Original entry on oeis.org

1, 2, 1, 3, 3, 3, 4, 6, 10, 6, 5, 10, 22, 23, 15, 6, 15, 40, 57, 62, 27, 7, 21, 65, 115, 165, 129, 63, 8, 28, 98, 205, 356, 385, 318, 120, 9, 36, 140, 336, 676, 914, 1005, 676, 252, 10, 45, 192, 518, 1176, 1885, 2524, 2334, 1524, 495, 11, 55, 255, 762, 1918, 3528, 5495, 6319, 5607, 3261, 1023
Offset: 1

Views

Author

Gus Wiseman, May 03 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.

Examples

			Triangle begins:
1
2    1
3    3    3
4    6   10    6
5   10   22   23   15
6   15   40   57   62   27
7   21   65  115  165  129   63
8   28   98  205  356  385  318  120
9   36  140  336  676  914 1005  676  252
The a(4,3) = 10 multisets: (112), (113), (122), (123), (124), (133), (134), (223), (233), (234).
The a(5,4) = 23 multisets:
(1112), (1222),
(1113), (1123), (1223), (1233), (1333), (2223), (2333),
(1124), (1134), (1224), (1234), (1244), (1334), (1344), (2234), (2334), (2344),
(1235), (1245), (1345), (2345).

Links

Crossrefs

Row sums are A303976.
Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.

Programs

  • Mathematica
    allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Length],{n,10}]
  • PARI
    T(n,k)={sumdiv(k, d, moebius(k/d)*sum(i=1, d, binomial(d-1, i-1)*binomial(n-k+i, i)))} \\ Andrew Howroyd, Sep 18 2018

Formula

T(n,k) = Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018

Extensions

Terms a(56) and beyond from Andrew Howroyd, Sep 18 2018

A303976 Number of different aperiodic multisets that fit within some normal multiset of size n.

Original entry on oeis.org

1, 3, 9, 26, 75, 207, 565, 1518, 4044, 10703, 28234, 74277, 195103, 511902, 1342147, 3517239, 9214412, 24134528, 63204417, 165505811, 433361425, 1134664831, 2970787794, 7777975396, 20363634815, 53313819160, 139579420528, 365427311171, 956707667616, 2504704955181
Offset: 1

Views

Author

Gus Wiseman, May 03 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.

Examples

			The a(4) = 26 aperiodic multisets:
(1), (2), (3), (4),
(12), (13), (14), (23), (24), (34),
(112), (113), (122), (123), (124), (133), (134), (223), (233), (234),
(1112), (1123), (1222), (1223), (1233), (1234).

Links

Crossrefs

Row sums of A303974.
Cf. A000740, A000837, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.

Programs

  • Mathematica
    allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&]],{n,10}]
  • PARI
    seq(n)={Vec(sum(d=1, n, moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x))} \\ Andrew Howroyd, Feb 04 2021

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018
G.f.: Sum_{d>=1} mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)). - Andrew Howroyd, Feb 04 2021

Extensions

Terms a(13) and beyond from Andrew Howroyd, Sep 18 2018

A303552 Number of periodic multisets of compositions of total weight n.

Original entry on oeis.org

0, 1, 1, 3, 1, 9, 1, 18, 7, 44, 1, 119, 1, 246, 48, 585, 1, 1470, 1, 3248, 250, 7535, 1, 18114, 42, 40593, 1373, 93726, 1, 218665, 1, 493735, 7539, 1127981, 285, 2587962, 1, 5841445, 40597, 13244166, 1, 30047413, 1, 67604050, 216745, 152258273, 1, 342747130
Offset: 1

Views

Author

Gus Wiseman, Apr 26 2018

Keywords

Comments

A multiset is periodic if its multiplicities have a common divisor greater than 1.

Examples

			The a(6) = 9 periodic multisets of compositions are:
{1,1,1,1,1,1},
{1,1,2,2}, {1,1,11,11},
{2,2,2}, {11,11,11},
{3,3}, {21,21}, {12,12}, {111,111}.

Crossrefs

Cf. A000740, A000837, A007716, A007916, A034691, A100953, A255906, A269134, A301700, A303386, A303431, A303547, A303551.

Programs

  • Mathematica
    nn=60;
ser=Product[1/(1-x^n)^2^(n-1),{n,nn}]
Table[SeriesCoefficient[ser,{x,0,n}]-Sum[MoebiusMu[d]*SeriesCoefficient[ser,{x,0,n/d}],{d,Divisors[n]}],{n,1,nn}]

A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 11, 8, 1, 10, 21, 27, 16, 1, 12, 38, 61, 63, 32, 1, 18, 57, 120, 162, 143, 64, 1, 22, 87, 205, 347, 409, 319, 128, 1, 28, 122, 333, 651, 950, 1000, 703, 256, 1, 32, 164, 506, 1132, 1926, 2504, 2391, 1535, 512, 1, 42, 217, 734, 1840
Offset: 1

Views

Author

Gus Wiseman, May 15 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers, and is aperiodic if its multiplicities are relatively prime.

Examples

			Triangle begins:
1
1    2
1    4    4
1    6   11    8
1   10   21   27   16
1   12   38   61   63   32
1   18   57  120  162  143   64
1   22   87  205  347  409  319  128
The a(4,3) = 11 multisets are (3), (13), (23), (113), (123), (133), (223), (233), (1123), (1223), (1233).

Links

Crossrefs

Row sums are A303976.
Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945, A303974.

Programs

  • Mathematica
    allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Max],{n,10}]
  • PARI
    T(n,k) = sum(j=1, n, sumdiv(j, d, sum(i=max(1, j+k-n), d, moebius(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1)))) \\ Andrew Howroyd, Jan 20 2023

Formula

T(n,k) = Sum_{j=1..n} Sum_{d|j} Sum_{i=max(1, j+k-n)..d} mu(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1). - Andrew Howroyd, Jan 20 2023
Showing 1-5 of 5 results.

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