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 1-10 of 10 results.

A291636 Matula-Goebel numbers of lone-child-avoiding rooted trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 133, 152, 172, 196, 212, 214, 224, 256, 262, 266, 301, 304, 326, 343, 344, 361, 371, 392, 424, 428, 448, 454, 512, 524, 526, 532, 602, 608, 622, 652, 686, 688, 722, 742, 749, 766, 784, 817
Offset: 1

Views

Author

Gus Wiseman, Aug 28 2017

Keywords

Comments

We say that a rooted tree is lone-child-avoiding if no vertex has exactly one child.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
An alternative definition: n is in the sequence iff n is 1 or the product of two or more not necessarily distinct prime numbers whose prime indices already belong to the sequence. For example, 14 is in the sequence because 14 = prime(1) * prime(4) and 1 and 4 both already belong to the sequence.

Examples

			The sequence of all lone-child-avoiding rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(o(oo)))

Links

Crossrefs

These trees are counted by A001678.
The case with more than two branches is A331490.
Unlabeled rooted trees are counted by A000081.
Topologically series-reduced rooted trees are counted by A001679.
Labeled lone-child-avoiding rooted trees are counted by A060356.
Labeled lone-child-avoiding unrooted trees are counted by A108919.
MG numbers of singleton-reduced rooted trees are A330943.
MG numbers of topologically series-reduced rooted trees are A331489.
Cf. A007097, A061775, A109082, A109129, A111299, A196050, A198518, A276625, A291441, A291442, A331488.

Programs

  • Mathematica
    nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[nn],srQ]

Extensions

Updated with corrected terminology by Gus Wiseman, Jan 20 2020

A330945 Numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84, 86, 87
Offset: 1

Views

Author

Gus Wiseman, Jan 13 2020

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 terms together with their prime indices of prime indices begins:
   2: {{}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
  10: {{},{2}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  28: {{},{},{1,1}}
  29: {{1,3}}

Crossrefs

Complement of A076610 (products of primes of prime index).
Numbers n such that A330944(n) > 0.
The restriction to odd terms is A330946.
The restriction to nonprimes is A330948.
The number of prime prime indices is given by A257994.
The number of nonprime prime indices is given by A330944.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320633, A330943, A330947, A330949.

Programs

  • Mathematica
    Select[Range[100],!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A331386 Numbers with at least one prime prime index.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 15, 17, 18, 20, 21, 22, 24, 25, 27, 30, 31, 33, 34, 35, 36, 39, 40, 41, 42, 44, 45, 48, 50, 51, 54, 55, 57, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 72, 75, 77, 78, 80, 81, 82, 83, 84, 85, 87, 88, 90, 93, 95, 96, 99, 100, 102, 105, 108
Offset: 1

Views

Author

Gus Wiseman, Jan 17 2020

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.
The asymptotic density of this sequence is 1 - Product_{p in A006450} (1 - 1/p) = 1 - 1/(Sum_{n>=1} 1/A076610(n)) > 2/3. - Amiram Eldar, Feb 02 2021

Examples

			The sequence of terms together with their prime indices begins:
    3: {2}
    5: {3}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   30: {1,2,3}
   31: {11}
   33: {2,5}
   34: {1,7}

Links

Crossrefs

Complement of A320628.
Positions of terms > 0 in A257994.
Positions of terms > 1 in A295665.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The number of nonprime prime indices is given by A330944.
Cf. A000040, A000720, A001222, A018252, A056239, A076610, A112798, A302242, A320633, A330943, A330944, A330947, A330949.

Programs

Formula

A257994(a(n)) > 0.

A330951 Number of singleton-reduced unlabeled rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 24, 52, 119, 272, 635, 1499, 3577, 8614, 20903, 51076, 125565, 310302, 770536, 1921440, 4809851, 12081986, 30445041, 76938794, 194950040, 495174037, 1260576786, 3215772264, 8219437433, 21046602265, 53982543827, 138678541693, 356785641107
Offset: 1

Views

Author

Gus Wiseman, Jan 15 2020

Keywords

Comments

A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.

Examples

			The a(1) = 1 through a(6) = 11 trees:
  o  (o)  (oo)  (ooo)   (oooo)    (ooooo)
                ((oo))  ((ooo))   ((oooo))
                (o(o))  (o(oo))   (o(ooo))
                        (oo(o))   (oo(oo))
                        ((o(o)))  (ooo(o))
                                  ((o)(oo))
                                  ((o(oo)))
                                  ((oo(o)))
                                  (o((oo)))
                                  (o(o)(o))
                                  (o(o(o)))

Links

Crossrefs

The Matula-Goebel numbers of these trees are given by A330943.
The series-reduced case is A001678.
Unlabeled rooted trees are counted by A000081.
Singleton-reduced phylogenetic trees are A000311.
Cf. A000669, A003238, A004111, A324694.

Programs

  • Mathematica
    urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],FreeQ[#,q:{__List}/;Times@@Length/@q==1]&]],{n,10}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n)); v[1]=1; for(n=1, #v-1, v[n+1] = EulerT(v[1..n])[n] - EulerT(Vec(x^2*Ser(v[1..n-1])/(1+x), -n))[n]); v} \\ Andrew Howroyd, Dec 10 2020

Formula

G.f.: A(x) satisfies A(x) = x + x*exp(Sum_{k>=1} A(x^k)/k) - x*exp(Sum_{k>=1} x^k*A(x^k)/(1 + x^k)/k). - Andrew Howroyd, Dec 10 2020
a(n) ~ c * d^n / n^(3/2), where d = 2.69474016697407303512228736537683134987637576... and c = 0.41800971384719166056172258174139385922545... - Vaclav Kotesovec, Nov 16 2021

Extensions

Terms a(19) and beyond from Andrew Howroyd, Dec 10 2020

A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 10, 20, 36, 70, 134, 263, 513, 1022, 2030, 4076, 8203, 16614, 33738, 68833, 140796, 288989, 594621, 1226781, 2536532, 5256303, 10913196, 22700682, 47299699, 98714362, 206323140, 431847121, 905074333, 1899247187, 3990145833, 8392281473
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2020

Keywords

Comments

Also the number of lone-child-avoiding rooted trees with n vertices and more than two branches.

Examples

			The a(4) = 1 through a(9) = 10 trees:
  (ooo)  (oooo)  (ooooo)   (oooooo)   (ooooooo)    (oooooooo)
                 (oo(oo))  (oo(ooo))  (oo(oooo))   (oo(ooooo))
                           (ooo(oo))  (ooo(ooo))   (ooo(oooo))
                                      (oooo(oo))   (oooo(ooo))
                                      (o(oo)(oo))  (ooooo(oo))
                                      (oo(o(oo)))  (o(oo)(ooo))
                                                   (oo(o(ooo)))
                                                   (oo(oo)(oo))
                                                   (oo(oo(oo)))
                                                   (ooo(o(oo)))

Links

Crossrefs

The not necessarily lone-child-avoiding version is A331233.
The Matula-Goebel numbers of these trees are listed by A331490.
A000081 counts unlabeled rooted trees.
A001678 counts lone-child-avoiding rooted trees.
A001679 counts topologically series-reduced rooted trees.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A331489 lists Matula-Goebel numbers of series-reduced rooted trees.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

Programs

  • Mathematica
    urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]>2&&FreeQ[#,{_}]&]],{n,10}]

Formula

For n > 1, a(n) = A001679(n) - A001678(n).

Extensions

a(37)-a(38) from Jinyuan Wang, Jun 26 2020
Terminology corrected (lone-child-avoiding, not series-reduced) by Gus Wiseman, May 10 2021

A330948 Nonprime numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106
Offset: 1

Views

Author

Gus Wiseman, Jan 13 2020

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 terms together with their prime indices of prime indices begins:
   4: {{},{}}
   6: {{},{1}}
   8: {{},{},{}}
  10: {{},{2}}
  12: {{},{},{1}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  22: {{},{3}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}
  34: {{},{4}}
  35: {{2},{1,1}}
  36: {{},{},{1},{1}}
  38: {{},{1,1,1}}

Crossrefs

Complement in A330945 of A000040.
Complement in A018252 of A076610.
The restriction to odd terms is A330949.
Nonprime numbers n such that A330944(n) > 0.
Taking odds instead of nonprimes gives A330946.
The number of prime prime indices is given by A257994.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000720, A001222, A056239, A112798, A302242, A320629, A320633, A330943, A330947.

Programs

  • Mathematica
    Select[Range[100],!PrimeQ[#]&&!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A331490 Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).

Original entry on oeis.org

8, 16, 28, 32, 56, 64, 76, 98, 112, 128, 152, 172, 196, 212, 224, 256, 266, 304, 343, 344, 392, 424, 428, 448, 512, 524, 532, 602, 608, 652, 686, 688, 722, 742, 784, 848, 856, 896, 908, 931, 1024, 1048, 1052, 1064, 1204, 1216, 1244, 1304, 1372, 1376, 1444
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2020

Keywords

Comments

We say that a rooted tree is (topologically) series-reduced if no vertex has degree 2.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.

Examples

			The sequence of all series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:
    8: (ooo)
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   98: (o(oo)(oo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
  212: (oo(oooo))
  224: (ooooo(oo))
  256: (oooooooo)
  266: (o(oo)(ooo))
  304: (oooo(ooo))
  343: ((oo)(oo)(oo))
  344: (ooo(o(oo)))

Links

Crossrefs

These trees are counted by A331488.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees are counted by A001678.
Topologically series-reduced rooted trees are counted by A001679.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Matula-Goebel numbers of series-reduced rooted trees are A331489.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[1000],PrimeOmega[#]>2&&srQ[#]&]

A330946 Odd numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

7, 13, 19, 21, 23, 29, 35, 37, 39, 43, 47, 49, 53, 57, 61, 63, 65, 69, 71, 73, 77, 79, 87, 89, 91, 95, 97, 101, 103, 105, 107, 111, 113, 115, 117, 119, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 159, 161, 163, 167, 169, 171, 173, 175, 181, 183, 185
Offset: 1

Views

Author

Gus Wiseman, Jan 13 2020

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.
Also MM-numbers of multiset partitions whose parts not all singletons (see example).

Examples

			The sequence of terms together with their prime indices of prime indices begins:
   7: {{1,1}}
  13: {{1,2}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  29: {{1,3}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  43: {{1,4}}
  47: {{2,3}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}
  61: {{1,2,2}}
  63: {{1},{1},{1,1}}
  65: {{2},{1,2}}
  69: {{1},{2,2}}
  71: {{1,1,3}}
  73: {{2,4}}

Crossrefs

Odd numbers n such that A330944(n) > 0.
Including even numbers gives A330945.
The restriction to nonprimes is A330949.
Taking nonprimes instead of odds gives A330947.
The number of prime prime indices is given by A257994.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320629, A330943, A330947, A330948.

Programs

  • Mathematica
    Select[Range[1,100,2],!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A331489 Matula-Goebel numbers of topologically series-reduced rooted trees.

Original entry on oeis.org

1, 2, 7, 8, 16, 19, 28, 32, 43, 53, 56, 64, 76, 98, 107, 112, 128, 131, 152, 163, 172, 196, 212, 224, 227, 256, 263, 266, 304, 311, 343, 344, 383, 392, 424, 428, 443, 448, 512, 521, 524, 532, 577, 602, 608, 613, 652, 686, 688, 719, 722, 742, 751, 784, 848, 856
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2020

Keywords

Comments

We say that a rooted tree is topologically series-reduced if no vertex (including the root) has degree 2.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.

Examples

			The sequence of all topologically series-reduced rooted trees together with their Matula-Goebel numbers begins:
    1: o
    2: (o)
    7: ((oo))
    8: (ooo)
   16: (oooo)
   19: ((ooo))
   28: (oo(oo))
   32: (ooooo)
   43: ((o(oo)))
   53: ((oooo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   98: (o(oo)(oo))
  107: ((oo(oo)))
  112: (oooo(oo))
  128: (ooooooo)
  131: ((ooooo))
  152: (ooo(ooo))
  163: ((o(ooo)))

Links

Crossrefs

Unlabeled rooted trees are counted by A000081.
Topologically series-reduced trees are counted by A000014.
Topologically series-reduced rooted trees are counted by A001679.
Labeled topologically series-reduced trees are counted by A005512.
Labeled topologically series-reduced rooted trees are counted by A060313.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000669, A001678, A007097, A007821, A060356, A061775, A109082, A109129, A196050, A254382, A276625, A330943, A331490.

Programs

  • Mathematica
    nn=1000;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[nn],PrimeOmega[#]!=2&&And@@srQ/@primeMS[#]&]

A330949 Odd nonprime numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

21, 35, 39, 49, 57, 63, 65, 69, 77, 87, 91, 95, 105, 111, 115, 117, 119, 129, 133, 141, 143, 145, 147, 159, 161, 169, 171, 175, 183, 185, 189, 195, 203, 207, 209, 213, 215, 217, 219, 221, 231, 235, 237, 245, 247, 253, 259, 261, 265, 267, 273, 285, 287, 291
Offset: 1

Views

Author

Gus Wiseman, Jan 14 2020

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.
Also MM-numbers of multiset partitions with at least two parts, not all of which are singletons (see example).

Examples

			The sequence of terms together with their prime indices of prime indices begins:
   21: {{1},{1,1}}
   35: {{2},{1,1}}
   39: {{1},{1,2}}
   49: {{1,1},{1,1}}
   57: {{1},{1,1,1}}
   63: {{1},{1},{1,1}}
   65: {{2},{1,2}}
   69: {{1},{2,2}}
   77: {{1,1},{3}}
   87: {{1},{1,3}}
   91: {{1,1},{1,2}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  111: {{1},{1,1,2}}
  115: {{2},{2,2}}
  117: {{1},{1},{1,2}}
  119: {{1,1},{4}}
  129: {{1},{1,4}}
  133: {{1,1},{1,1,1}}
  141: {{1},{2,3}}

Crossrefs

Complement of A106092 in A330945.
Including even numbers gives A330948.
Including primes gives A330946.
The number of prime prime indices is given by A257994.
The number of nonprime prime indices is given by A330944.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320629, A320633, A330943, A330947.

Programs

  • Mathematica
    Select[Range[1,100,2],!PrimeQ[#]&&!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]
Showing 1-10 of 10 results.

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