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.

A325660 Number of ones in the q-signature of n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, May 13 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)
Then a(n) is the number of factors of multiplicity one in the q-factorization of n.
Also the number of rooted trees appearing only once in the multiset of terminal subtrees of the rooted tree with Matula-Goebel number n.

Crossrefs

Cf. A001694, A055231, A056169, A056239, A112798, A118914, A124010.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324968.
q-factorization: A324922, A324923, A324924, A325614, A325661.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Count[Length/@Split[difac[n]],1],{n,100}]

A325662 Matula-Goebel numbers of regular rooted stars.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 16, 25, 27, 31, 32, 64, 81, 121, 125, 127, 128, 243, 256, 512, 625, 709, 729, 961, 1024, 1331, 2048, 2187, 3125, 4096, 5381, 6561, 8192, 14641, 15625, 16129, 16384, 19683, 29791, 32768, 52711, 59049, 65536, 78125, 131072, 161051
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

Powers of members of A007097.
A regular rooted star is a rooted tree whose branches are all rooted paths of equal length.
The number of terms <= 10^k, k=0,1,2,...: 1, 7, 15, 26, 35, 46, 56, 67, 76, 87, 98, 109, 121, 131, 142, 154, 163, 175, 185, 198, 208, 220, 231, 241, 254, 265, 275, etc. - Robert G. Wilson v, May 13 2019

Examples

			The sequence of regular rooted stars together with their Matula-Goebel numbers begins:
    1: o
    2: (o)
    3: ((o))
    4: (oo)
    5: (((o)))
    8: (ooo)
    9: ((o)(o))
   11: ((((o))))
   16: (oooo)
   25: (((o))((o)))
   27: ((o)(o)(o))
   31: (((((o)))))
   32: (ooooo)
   64: (oooooo)
   81: ((o)(o)(o)(o))
  121: ((((o)))(((o))))
  125: (((o))((o))((o)))
  127: ((((((o))))))
  128: (ooooooo)

Links

Crossrefs

Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325663.

Programs

  • Mathematica
    rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100],#==1||PrimePowerQ[#]&&rpQ[FactorInteger[#][[1,1]]]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[emax]; s = Union[s, s1], {k, 2, Length[ps]}]; s]; seq[10] (* Amiram Eldar, Jul 26 2024 *)

Formula

Sum_{n>=1} 1/a(n) = 1 + Product_{k>=1} 1/(A007097(k)-1) = 2.8928887669834086909... - Amiram Eldar, Jul 26 2024

A325663 Matula-Goebel numbers of not necessarily regular rooted stars.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 36, 40, 44, 45, 48, 50, 54, 55, 60, 62, 64, 66, 72, 75, 80, 81, 88, 90, 93, 96, 99, 100, 108, 110, 120, 121, 124, 125, 127, 128, 132, 135, 144, 150, 155, 160, 162, 165, 176
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

Products of members of A007097.
A rooted star is a rooted tree whose branches are all rooted paths.

Examples

			The sequence of rooted stars together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  15: ((o)((o)))
  16: (oooo)
  18: (o(o)(o))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  30: (o(o)((o)))

Links

Crossrefs

Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325662.

Programs

  • Mathematica
    rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100],And@@rpQ/@First/@FactorInteger[#]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1, s2}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[0, emax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= pmax &]; s = Union[s, s2], {k, 2, Length[ps]}]; s]; seq[7] (* Amiram Eldar, Jul 26 2024 *)

Formula

Sum_{n>=1} 1/a(n) = Product_{k>=1} A007097(k)/(A007097(k)-1) = 4.30328607286382284593... . - Amiram Eldar, Jul 26 2024

A325697 Number of rooted trees with n vertices with no proper terminal subtree appearing at only one position.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 11, 13, 27, 30, 69, 76, 168
Offset: 1

Views

Author

Gus Wiseman, May 17 2019

Keywords

Comments

The Matula-Goebel numbers of these trees are given by A325661.

Examples

			The a(4) = 1 through a(9) = 11 rooted trees:
  (ooo)  (oooo)    (ooooo)    (oooooo)      (ooooooo)      (oooooooo)
         ((o)(o))  (o(o)(o))  ((oo)(oo))    (o(oo)(oo))    ((ooo)(ooo))
                              (oo(o)(o))    (ooo(o)(o))    (oo(oo)(oo))
                              ((o)(o)(o))   (o(o)(o)(o))   (oooo(o)(o))
                              (((o))((o)))  (o((o))((o)))  (oo(o)(o)(o))
                                                           (((oo))((oo)))
                                                           ((o)(o)(o)(o))
                                                           ((o(o))(o(o)))
                                                           (oo((o))((o)))
                                                           ((o)((o))((o)))
                                                           ((((o)))(((o))))

Crossrefs

Cf. A000081, A001694, A004111, A290689, A306844, A317713, A324936, A324971, A325661.

Programs

  • Mathematica
    urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],!MemberQ[Length/@Split[Sort[Extract[#,Most[Position[#,_List]]]]],1]&]],{n,15}]
Showing 1-4 of 4 results.

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

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