A298536 -id:A298536

The OEIS is supported by the many generous donors to the OEIS Foundation.

Search: a298536 -id:a298536

Displaying 1-5 of 5 results found.

page 1

A303431

Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.

+10
40

1, 2, 3, 5, 6, 10, 11, 12, 13, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 37, 39, 40, 41, 44, 45, 47, 48, 50, 52, 54, 55, 58, 60, 61, 62, 65, 66, 71, 72, 74, 75, 78, 79, 80, 82, 87, 88, 89, 90, 93, 94, 96, 99, 101, 104, 108, 109, 110, 111, 113, 116, 117, 120, 122

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A positive integer is an aperiodic tree number iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) and all of its prime indices are also aperiodic tree numbers, where a prime index of n is a number m such that prime(m) divides n.

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

Sequence of aperiodic rooted trees begins:

01 o

02 (o)

03 ((o))

05 (((o)))

06 (o(o))

10 (o((o)))

11 ((((o))))

12 (oo(o))

13 ((o(o)))

15 ((o)((o)))

18 (o(o)(o))

20 (oo((o)))

22 (o(((o))))

24 (ooo(o))

26 (o(o(o)))

29 ((o((o))))

30 (o(o)((o)))

31 (((((o)))))

33 ((o)(((o))))

MATHEMATICA

zapQ[1]:=True; zapQ[n_]:=And[GCD@@FactorInteger[n][[All, 2]]===1, And@@zapQ/@PrimePi/@FactorInteger[n][[All, 1]]];

Select[Range[100], zapQ]

CROSSREFS

Cf. A000081, A000740, A000837, A007097, A007916, A052409, A052410, A061775, A111299, A214577, A275024, A276625, A290760, A290822, A291442, A298533, A298536, A301700, A302242, A303386.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 23 2018

STATUS

approved

A298535

Number of unlabeled rooted trees with n vertices such that every branch of the root has a different number of leaves.

+10
5

1, 1, 1, 2, 5, 13, 32, 80, 200, 511, 1323, 3471, 9183, 24491, 65715, 177363, 481135, 1311340, 3589023, 9860254, 27181835, 75165194, 208439742, 579522977, 1615093755, 4511122964, 12625881944, 35405197065, 99459085125, 279861792874, 788712430532, 2226015529592

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..500

MATHEMATICA

rut[n_]:=rut[n]=If[n===1, {{}}, Join@@Function[c, Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];

Table[Length[Select[rut[n], UnsameQ@@(Count[#, {}, {0, Infinity}]&/@#)&]], {n, 15}]

PROG

(PARI) \\ here R is A055277 as vector of polynomials

R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1 + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};

seq(n) = {my(M=Mat(apply(p->Colrev(p, n), R(n-1)))); Vec(prod(i=2, #M, 1 + x*Ser(M[i, ])))} \\ Andrew Howroyd, May 20 2018

CROSSREFS

Cf. A000081, A003238, A004111, A032305, A289079, A290689, A291443, A297791, A298422, A298533, A298536.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 20 2018

EXTENSIONS

Terms a(19) and beyond from Andrew Howroyd, May 20 2018

STATUS

approved

A298534

Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of leaves.

+10
4

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 53, 54, 55, 59, 60, 61, 62, 64, 66, 67, 71, 72, 73, 75, 79, 80, 81, 83, 88, 89, 90, 91, 93, 96, 97, 99, 100

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

Sequence of trees begins:

1 o

2 (o)

3 ((o))

4 (oo)

5 (((o)))

6 (o(o))

7 ((oo))

8 (ooo)

9 ((o)(o))

10 (o((o)))

11 ((((o))))

12 (oo(o))

13 ((o(o)))

15 ((o)((o)))

16 (oooo)

17 (((oo)))

18 (o(o)(o))

19 ((ooo))

20 (oo((o)))

22 (o(((o))))

23 (((o)(o)))

24 (ooo(o))

25 (((o))((o)))

27 ((o)(o)(o))

29 ((o((o))))

30 (o(o)((o)))

MATHEMATICA

nn=2000;

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

leafcount[n_]:=If[n===1, 1, With[{m=primeMS[n]}, If[Length[m]===1, leafcount[First[m]], Total[leafcount/@m]]]];

Select[Range[nn], SameQ@@leafcount/@primeMS[#]&]

CROSSREFS

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A290822, A291442, A298533, A298536.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 20 2018

STATUS

approved

A298538

Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of nodes.

+10
3

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 187

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..61.

EXAMPLE

Sequence of trees begins:

1 o

2 (o)

3 ((o))

4 (oo)

5 (((o)))

7 ((oo))

8 (ooo)

9 ((o)(o))

11 ((((o))))

13 ((o(o)))

16 (oooo)

17 (((oo)))

19 ((ooo))

23 (((o)(o)))

25 (((o))((o)))

27 ((o)(o)(o))

29 ((o((o))))

31 (((((o)))))

MATHEMATICA

nn=500;

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

MGweight[n_]:=If[n===1, 1, 1+Total[MGweight/@primeMS[n]]];

Select[Range[nn], SameQ@@MGweight/@primeMS[#]&]

CROSSREFS

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A291442, A297571, A298478, A298534, A298536, A298537.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 21 2018

STATUS

approved

A298540

Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of nodes.

+10
1

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..65.

EXAMPLE

Sequence of trees begins:

1 o

2 (o)

3 ((o))

5 (((o)))

6 (o(o))

7 ((oo))

10 (o((o)))

11 ((((o))))

13 ((o(o)))

14 (o(oo))

15 ((o)((o)))

17 (((oo)))

19 ((ooo))

21 ((o)(oo))

22 (o(((o))))

23 (((o)(o)))

26 (o(o(o)))

29 ((o((o))))

30 (o(o)((o)))

MATHEMATICA

nn=500;

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

MGweight[n_]:=If[n===1, 1, 1+Total[MGweight/@primeMS[n]]];

Select[Range[nn], UnsameQ@@MGweight/@primeMS[#]&]

CROSSREFS

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A291442, A297571, A298479, A298534, A298536, A298538, A298539.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 21 2018

STATUS

approved

page 1

Search completed in 0.011 seconds