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.

Previous Showing 11-17 of 17 results.

A325614 Unsorted q-signature of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 4, 2, 1, 1, 3, 2, 3, 1, 3, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 2, 1, 4, 1, 2, 2, 2, 3, 1, 1, 3, 3, 4, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 3, 1, 1, 3, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
11 = q(1) q(2) q(3) q(5)
50 = q(1)^3 q(2)^2 q(3)^2
360 = q(1)^6 q(2)^3 q(3)
Row n lists the nonzero multiplicities in the q-factorization of n, in order of q-index. For example, row 11 is (1,1,1,1) and row 360 is (6,3,1).

Examples

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  2 1
  2 1
  3
  2 2
  2 1 1
  1 1 1 1
  3 1
  2 1 1
  3 1
  2 2 1
  4
  2 1 1
  3 2
  3 1
  3 1 1

Crossrefs

Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325615, A325660.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length/@Split[difac[n]],{n,30}]

A324971 Number of rooted identity trees with n vertices whose non-leaf terminal subtrees are not all different.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 12, 31, 79, 192, 459, 1082, 2537, 5922, 13816, 32222, 75254, 176034, 412667, 969531, 2283278
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root.

Examples

			The a(6) = 1 through a(8) = 12 trees:
  ((o)((o)))  ((o)(o(o)))   (o(o)(o(o)))
              (o(o)((o)))   (((o))(o(o)))
              (((o)((o))))  (((o)(o(o))))
              ((o)(((o))))  ((o)((o(o))))
                            ((o)(o((o))))
                            ((o(o)((o))))
                            (o((o)((o))))
                            (o(o)(((o))))
                            ((((o)((o)))))
                            (((o))(((o))))
                            (((o)(((o)))))
                            ((o)((((o)))))

Crossrefs

The Matula-Goebel numbers of these trees are given by A324970.
Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935, A324936, A324979.

Programs

  • Mathematica
    rits[n_]:=Join@@Table[Select[Union[Sort/@Tuples[rits/@ptn]],UnsameQ@@#&],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[rits[n],!UnsameQ@@Cases[#,{},{0,Infinity}]&]],{n,10}]

A325615 Sorted q-signature of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 1, 2, 2, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 2, 1, 1, 3, 3, 3, 1, 4, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
11 = q(1) q(2) q(3) q(5)
50 = q(1)^3 q(2)^2 q(3)^2
360 = q(1)^6 q(2)^3 q(3)
Row n is the multiset of nonzero multiplicities in the q-factorization of n. For example, row 11 is (1,1,1,1) and row 360 is (1,3,6).

Examples

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  1 2
  1 2
  3
  2 2
  1 1 2
  1 1 1 1
  1 3
  1 1 2
  1 3
  1 2 2
  4
  1 1 2
  2 3
  1 3
  1 1 3

Crossrefs

Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325614, A325660.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Sort[Length/@Split[difac[n]]],{n,30}]

A324933 Denominator in the division of n by the product of prime indices of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 4, 3, 5, 1, 6, 2, 2, 1, 7, 2, 8, 3, 8, 5, 9, 1, 9, 3, 8, 1, 10, 1, 11, 1, 10, 7, 12, 1, 12, 4, 4, 3, 13, 4, 14, 5, 4, 9, 15, 1, 16, 9, 14, 3, 16, 4, 3, 1, 16, 5, 17, 1, 18, 11, 16, 1, 18, 5, 19, 7, 6, 6, 20, 1, 21, 6, 6, 2, 20, 2, 22, 3
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

Links

Crossrefs

The numerators are A324932.
Cf. A003963, A109129, A120383, A196050, A317713, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

Programs

  • Mathematica
    Table[n/Times@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]^k],{n,100}]//Denominator

A324932 Numerator in the division of n by the product of prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 5, 16, 17, 9, 19, 20, 21, 22, 23, 12, 25, 13, 27, 7, 29, 5, 31, 32, 33, 34, 35, 9, 37, 19, 13, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 26, 53, 27, 11, 14, 57, 29, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 23
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

Links

Crossrefs

The denominators are A324933.
Cf. A003963, A109129, A120383, A196050, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

Programs

  • Mathematica
    Table[n/Times@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]^k],{n,100}]//Numerator

A324979 Number of rooted trees with n vertices that are not identity trees but whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 29, 70, 168, 402, 959, 2284, 5434, 12923, 30727, 73055, 173678, 412830
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.

Examples

			The a(3) = 1 through a(6) = 12 trees:
  (oo)  (ooo)   (oooo)    (ooooo)
        ((oo))  ((ooo))   ((oooo))
                (o(oo))   (o(ooo))
                (oo(o))   (oo(oo))
                (((oo)))  (ooo(o))
                          (((ooo)))
                          ((o)(oo))
                          ((o(oo)))
                          ((oo(o)))
                          (o((oo)))
                          (oo((o)))
                          ((((oo))))

Crossrefs

The Matula-Goebel numbers of these trees are given by A324978.
Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935, A324936, A324970, A324971.

Programs

  • Mathematica
    rits[n_]:=Join@@Table[Union[Sort/@Tuples[rits/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[rits[n],And[UnsameQ@@Cases[#,{},{0,Infinity}],!And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}]]&]],{n,10}]

A325608 Numbers whose factorization into factors prime(i)/i does not have weakly decreasing nonzero multiplicities.

Original entry on oeis.org

147, 245, 294, 357, 490, 511, 539, 588, 595, 637, 681, 714, 735, 845, 847, 853, 867, 903, 980, 1022, 1029, 1043, 1078, 1083, 1135, 1176, 1183, 1190, 1239, 1241, 1267, 1274, 1309, 1362, 1421, 1428, 1445, 1470, 1505, 1519, 1547, 1553, 1563, 1617, 1631, 1690
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example, 147 = q(1)^5 q(2) q(4)^2 has multiplicities (5,1,2), which are not weakly decreasing, so 147 belongs to the sequence.

Crossrefs

Cf. A001222, A056239, A112798, A118914.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.
q-factorization: A324922, A324923, A324924, A324931, A325613, A325614, A325615, A325660, A325662.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[1000],!GreaterEqual@@Length/@Split[difac[#]]&]
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.