A298538 -id:A298538 - cp's OEIS frontend

A326534 MM-numbers of multiset partitions where every part has the same sum.

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, 191

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

First differs from A298538 in lacking 187.

These are numbers where each prime index has the same sum of prime indices. 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 multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of multiset partitions where every part has the same sum, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}
  35: {{2},{1,1}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A035470, A038041, A112798, A302242, A320324, A321455, A326518, A326533, A326535, A326536, A326537.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@primeMS/@primeMS[#]&]

A298537 Number of unlabeled rooted trees with n nodes such that every branch of the root has the same number of nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 25, 49, 127, 291, 766, 1843, 5003, 12487, 34151, 87983, 242088, 634848, 1763749, 4688677, 13085621, 35241441, 98752586, 268282856, 755353825, 2067175933, 5837592853, 16087674276, 45550942142, 126186554309, 358344530763, 997171512999

Offset: 1

Gus Wiseman, Jan 20 2018

nonn

			The a(5) = 6 trees: ((((o)))), (((oo))), ((o(o))), ((ooo)), ((o)(o)), (oooo).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000081, A003238, A004111, A032305, A289078, A289079, A290689, A291443, A297791, A298422, A298533, A298535, A298538, A298539.

r[n_]:=r[n]=If[n===1,1,Sum[Product[Binomial[r[x]+Count[ptn,x]-1,Count[ptn,x]],{x,Union[ptn]}],{ptn,IntegerPartitions[n-1]}]];
Table[If[n===1,1,Sum[Binomial[r[(n-1)/d]+d-1,d],{d,Divisors[n-1]}]],{n,40}]

a(n + 1) = Sum_{d|n} binomial(A000081(n/d) + d - 1, d).

A298540 Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of nodes.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 21 2018

nonn

			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)))

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

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[#]&]

cp's OEIS Frontend

A326534 MM-numbers of multiset partitions where every part has the same sum.

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A298537 Number of unlabeled rooted trees with n nodes such that every branch of the root has the same number of nodes.

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A298540 Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of nodes.

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Mathematica