A317712 -id:A317712 - cp's OEIS frontend

A317717 Uniform relatively prime tree numbers. Matula-Goebel numbers of uniform relatively prime rooted trees.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 22, 26, 29, 30, 31, 32, 33, 34, 35, 36, 38, 41, 42, 43, 47, 51, 53, 55, 58, 59, 62, 64, 66, 67, 70, 77, 78, 79, 82, 85, 86, 93, 94, 95, 100, 101, 102, 105, 106, 109, 110, 113, 114, 118, 119, 123, 127, 128

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform relatively prime tree number iff either n = 1 or n is a prime number whose prime index is a uniform relatively prime tree number, or n is a power of a squarefree number whose prime indices are relatively prime and are themselves uniform relatively prime tree numbers. A prime index of n is a number m such that prime(m) divides n.

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000081, A061775, A072774, A214577, A276625, A277098, A317588.

Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317712, A317718.

rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[200],rupQ]

A317711 Numbers that are not uniform tree numbers.

Original entry on oeis.org

12, 18, 20, 24, 28, 37, 40, 44, 45, 48, 50, 52, 54, 56, 60, 61, 63, 68, 71, 72, 74, 75, 76, 80, 84, 88, 89, 90, 92, 96, 98, 99, 104, 107, 108, 111, 112, 116, 117, 120, 122, 124, 126, 132, 135, 136, 140, 142, 144, 147, 148, 150, 152, 153, 156, 157, 160, 162

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform tree number iff either n = 1 or n is a power of a squarefree number whose prime indices are also uniform tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of non-uniform tree numbers together with their Matula-Goebel trees begins:
  12: (oo(o))
  18: (o(o)(o))
  20: (oo((o)))
  24: (ooo(o))
  28: (oo(oo))
  37: ((oo(o)))
  40: (ooo((o)))
  44: (oo(((o))))
  45: ((o)(o)((o)))
  48: (oooo(o))
  50: (o((o))((o)))
  52: (oo(o(o)))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  60: (oo(o)((o)))

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317590.

Cf. A317705, A317707, A317708, A317709, A317710 (complement), A317712, A317717, A317718.

rupQ[n_]:=Or[n==1,And[SameQ@@FactorInteger[n][[All,2]],And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[100],!rupQ[#]&]

A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 44, 70, 119, 189, 314, 506, 830, 1336, 2186, 3522, 5737, 9266, 15047, 24313, 39444, 63759, 103322, 167098, 270616, 437714, 708676, 1146390, 1855582, 3002017, 4858429, 7860454, 12720310, 20580764, 33303260, 53884144, 87190964

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

This is a weaker condition than achirality (cf. A167865).

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 1 through a(8) = 9 rooted trees:
  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)
               (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))
                        (oo(oo))  (oo(ooo))   (oo(oooo))
                                  (ooo(oo))   (ooo(ooo))
                                  ((oo)(oo))  (oooo(oo))
                                  (o(o(oo)))  (o(o(ooo)))
                                              (o(oo)(oo))
                                              (o(oo(oo)))
                                              (oo(o(oo)))

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

Cf. A001678, A002541, A003238, A010766, A070776, A014668, A126656, A167865, A317099, A317100, A317712, A320222, A320226, A320269.

saum[n_]:=Sum[If[DeleteCases[ptn,1]=={},1,saum[DeleteCases[ptn,1][[1]]]],{ptn,Select[IntegerPartitions[n-1],And[Length[#]!=1,SameQ@@DeleteCases[#,1]]&]}];
Array[saum,20]

seq(n)={my(v=vector(n)); v[1]=1; for(n=3, n, v[n] = 1 + sum(k=2, n-2, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018

a(1) = 1; a(2) = 0; a(n > 2) = 1 + Sum_{k = 2..n-2} floor((n-1)/k) * a(k).

A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 8, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Sep 16 2018

nonn

			The a(144) = 17 factorizations:
  (144),
  (2*72), (3*48), (4*36), (6*24), (8*18), (9*16), (12*12),
  (2*3*24), (2*4*18), (2*6*12), (2*8*9), (3*4*12), (3*6*8),
  (2*2*6*6), (2*3*4*6), (3*3*4*4).

Cf. A001055, A045778, A047966, A052409, A072774, A296132, A303386, A317710, A317712, A317717, A317718.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@Length/@Split[#]&]],{n,100}]

a(n) = Sum_{d|A052409(n)} A045778(n^(1/d)).

A320224 a(1) = 1; a(n > 1) = Sum_{k = 1..n-1} Sum_{d|k, d < k} a(d).

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 7, 10, 12, 16, 17, 25, 26, 33, 38, 48, 49, 65, 66, 84, 92, 109, 110, 142, 146, 172, 184, 219, 220, 274, 275, 323, 341, 390, 400, 484, 485, 551, 578, 669, 670, 792, 793, 904, 952, 1062, 1063, 1243, 1250, 1408, 1458, 1632, 1633, 1870, 1890

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002541, A003238, A010766, A014668, A126656, A167865, A316782, A317712, A320222, A320225.

sol:=[1]; for n in [2..56] do Append(~sol, &+[sol[d]*Floor((n-1)/d-1):d in [1..n-1]]); end for; sol; // Marius A. Burtea, Sep 07 2019

sau[n_]:=If[n==1,1,Sum[sau[d],{k,n-1},{d,Most[Divisors[k]]}]];
Table[sau[n],{n,60}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(k=1, n-1, v[k]*((n-1)\k - 1))); v} \\ Andrew Howroyd, Sep 07 2019

a(1) = 1; a(n > 1) = Sum_{d = 1..n-1} a(d) * floor((n-1)/d - 1).

G.f. A(x) satisfies A(x) = x + (x/(1 - x)) * Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019

A320225 a(1) = 1; a(n > 1) = Sum_{k = 1..n} Sum_{d|k, d < k} a(d).

Original entry on oeis.org

1, 1, 2, 4, 5, 9, 10, 16, 19, 26, 27, 44, 45, 57, 65, 87, 88, 120, 121, 158, 171, 200, 201, 278, 284, 331, 353, 426, 427, 536, 537, 646, 676, 766, 782, 982, 983, 1106, 1154, 1365, 1366, 1617, 1618, 1851, 1943, 2146, 2147, 2589, 2600, 2917, 3008, 3390, 3391

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002541, A003238, A010766, A014668, A126656, A167865, A316782, A317712, A320222, A320224.

sau[n_]:=If[n==1,1,Sum[sau[d],{k,n},{d,Most[Divisors[k]]}]];
Table[sau[n],{n,30}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A320225(n): return 1 if n == 1 else sum(A320225(d)*(n//d-1) for d in range(1,n)) # Chai Wah Wu, Jun 08 2022

a(1) = 1; a(n > 1) = Sum_{d = 1..n-1} a(d) * floor(n/d-1).

G.f. A(x) satisfies A(x) = x + (1/(1 - x)) * Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019

A320226 Number of integer partitions of n whose non-1 parts are all equal.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 13, 15, 18, 19, 24, 25, 28, 31, 35, 36, 41, 42, 47, 50, 53, 54, 61, 63, 66, 69, 74, 75, 82, 83, 88, 91, 94, 97, 105, 106, 109, 112, 119, 120, 127, 128, 133, 138, 141, 142, 151, 153, 158, 161, 166, 167, 174, 177, 184, 187, 190, 191, 202

Offset: 0

Gus Wiseman, Oct 07 2018

nonn

			The integer partitions whose non-1 parts are all equal:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (51)      (331)      (71)
                    (211)   (311)    (222)     (511)      (611)
                    (1111)  (2111)   (411)     (2221)     (2222)
                            (11111)  (2211)    (4111)     (3311)
                                     (3111)    (22111)    (5111)
                                     (21111)   (31111)    (22211)
                                     (111111)  (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Jonathan Bloom, Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.

Cf. A002541, A003238, A070776, A126656, A316782, A317712, A320224, A320225.

Table[Length[Select[IntegerPartitions[n],SameQ@@DeleteCases[#,1]&]],{n,30}]

a(n > 1) = A002541(n - 1) + 1.

A317719 Numbers that are not powerful tree numbers.

Original entry on oeis.org

6, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 26, 28, 29, 30, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a powerful tree number iff either n = 1 or n is a prime number whose prime index is a powerful tree number, or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all powerful tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of numbers that are not powerful tree numbers together with their Matula-Goebel trees begins:
   6: (o(o))
  10: (o((o)))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Complement of A318612.
Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.
Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718.

powgoQ[n_]:=Or[n==1,If[PrimeQ[n],powgoQ[PrimePi[n]],And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!powgoQ[#]&]

A320271 Number of unlabeled semi-binary rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 26, 46, 72, 124, 196, 329, 525, 871, 1396, 2293, 3689, 6028, 9717, 15817, 25534, 41475, 67009, 108680, 175689, 284698, 460387, 745610, 1205997, 1952478, 3158475, 5112349, 8270824, 13385466, 21656290, 35045445, 56701735, 91753208

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

An unlabeled rooted tree is semi-binary if all out-degrees are <= 2. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

			The a(1) = 1 through a(7) = 17 semi-binary rooted trees:
  o  (o)  (oo)   ((oo))   (o(oo))    ((o(oo)))    ((oo)(oo))
          ((o))  (o(o))   (((oo)))   (o((oo)))    (o(o(oo)))
                 (((o)))  ((o)(o))   (o(o(o)))    (((o(oo))))
                          ((o(o)))   ((((oo))))   ((o((oo))))
                          (o((o)))   (((o)(o)))   ((o(o(o))))
                          ((((o))))  (((o(o))))   (o(((oo))))
                                     ((o((o))))   (o((o)(o)))
                                     (o(((o))))   (o((o(o))))
                                     (((((o)))))  (o(o((o))))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  ((((o(o)))))
                                                  (((o))((o)))
                                                  (((o((o)))))
                                                  ((o(((o)))))
                                                  (o((((o)))))
                                                  ((((((o))))))

Cf. A001190, A003238, A111299, A126656, A292050, A298204, A301345, A317712, A320222, A320230, A320270.

crb[n_]:=Switch[n,1,1,2,1,3,2,?EvenQ,crb[n-1]+crb[n-2],?OddQ,crb[n-1]+crb[n-2]+crb[(n-1)/2]]
Array[crb,20]

       a(1) = 1,
       a(2) = 1,
       a(3) = 2,
  a(n even) = a(n-1) + a(n-2),
   a(n odd) = a(n-1) + a(n-2) + a((n-1)/2).

A317720 Numbers that are not uniform relatively prime tree numbers.

Original entry on oeis.org

9, 12, 18, 20, 21, 23, 24, 25, 27, 28, 37, 39, 40, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 60, 61, 63, 65, 68, 69, 71, 72, 73, 74, 75, 76, 80, 81, 83, 84, 87, 88, 89, 90, 91, 92, 96, 97, 98, 99, 103, 104, 107, 108, 111, 112, 115, 116, 117, 120, 121, 122, 124

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform relatively prime tree number iff either n = 1 or n is a prime number whose prime index is a uniform relatively prime tree number, or n is a power of a squarefree number whose prime indices are relatively prime and are themselves uniform relatively prime tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of non-uniform tree numbers together with their Matula-Goebel trees begins:
   9: ((o)(o))
  12: (oo(o))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  23: (((o)(o)))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  37: ((oo(o)))
  39: ((o)(o(o)))
  40: (ooo((o)))
  44: (oo(((o))))
  45: ((o)(o)((o)))

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317590.

Cf. A317705, A317707, A317708, A317709, A317710, A317712, A317717, A317718.

rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[200],!rupQ[#]&]

cp's OEIS Frontend

A317717 Uniform relatively prime tree numbers. Matula-Goebel numbers of uniform relatively prime rooted trees.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A317711 Numbers that are not uniform tree numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A320224 a(1) = 1; a(n > 1) = Sum_{k = 1..n-1} Sum_{d|k, d < k} a(d).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A320225 a(1) = 1; a(n > 1) = Sum_{k = 1..n} Sum_{d|k, d < k} a(d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A320226 Number of integer partitions of n whose non-1 parts are all equal.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A317719 Numbers that are not powerful tree numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica