A303431 -id:A303431 - cp's OEIS frontend

A304360 Lexicographically earliest infinite sequence of numbers m > 1 with the property that none of the prime indices of m are in the sequence.

2, 4, 5, 8, 10, 13, 16, 17, 20, 23, 25, 26, 31, 32, 34, 37, 40, 43, 46, 47, 50, 52, 61, 62, 64, 65, 67, 68, 73, 74, 79, 80, 85, 86, 89, 92, 94, 100, 103, 104, 107, 109, 113, 115, 122, 124, 125, 128, 130, 134, 136, 137, 146, 148, 149, 151, 155, 158, 160, 163

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

A self-describing sequence.
The prime indices of m are the numbers k such that prime(k) divides m.
The sequence is monotonically increasing, since once a number is rejected it stays rejected. Sequence is closed under multiplication for a similar reason. - N. J. A. Sloane, Aug 26 2018

			After the initial term 2, the next term cannot be 3 because 3 has prime index 2, and 2 is already in the sequence. The next term could be 10, which has prime indices 1 and 3, but 4 (with prime index 1) is smaller. So a(2) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000002, A001462, A079000, A079254, A214577, A276625, A277098, A280996, A303431.

For first differences see A317963, for primes see A317964.

A:= NULL:
P:= {}:
for n  from 2 to 1000 do
  pn:= numtheory:-factorset(n);
  if pn intersect P = {} then
    A:= A, n;
    P:= P union {ithprime(n)};
  fi
od:
A; # Robert Israel, Aug 26 2018

gaQ[n_]:=Or[n==0,And@@Cases[FactorInteger[n],{p_,k_}:>!gaQ[PrimePi[p]]]];
Select[Range[100],gaQ]

Added "infinite" to definition. - N. J. A. Sloane, Sep 28 2019

A303386 Number of aperiodic factorizations of n > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 2, 4, 1, 5, 1, 6, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 7, 1, 5, 1, 7, 5

Offset: 2

Gus Wiseman, Apr 23 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

			The a(36) = 7 aperiodic factorizations are (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), and (36). Missing from this list are (2*2*3*3) and (6*6).

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A000740, a(2^n) = A000837(n), A001055, A007716, A007916, A045778, A052409, A052410, A100953, A162247, A275024, A275870, A281113, A281116, A301700, A302242, A303431.

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],GCD@@Length/@Split[#]===1&]],{n,2,100}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A303386(n) = if(1==n,n,my(r); sumdiv(A052409(n),d, ispower(n,d,&r); moebius(d)*A001055(r))); \\ Antti Karttunen, Sep 25 2018

a(n) = Sum_{d|A052409(n)} mu(d) * A001055(n^(1/d)), where mu = A008683.

More terms from Antti Karttunen, Sep 25 2018

A301700 Number of aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 21, 52, 120, 290, 697, 1713, 4200, 10446, 26053, 65473, 165257, 419357, 1068239, 2732509, 7013242, 18059960, 46641983, 120790324, 313593621, 816046050, 2128101601, 5560829666, 14557746453, 38177226541, 100281484375, 263815322761, 695027102020

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

An unlabeled rooted tree is aperiodic if the multiset of branches of the root is an aperiodic multiset, meaning it has relatively prime multiplicities, and each branch is also aperiodic.

			The a(6) = 10 aperiodic trees are (((((o))))), (((o(o)))), ((o((o)))), ((oo(o))), (o(((o)))), (o(o(o))), ((o)((o))), (oo((o))), (o(o)(o)), (ooo(o)).

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

Cf. A000081, A000740, A000837, A001678, A003238, A004111, A007716, A007916, A100953, A276625, A284639, A290689, A298422, A303386, A303431.

arut[n_]:=arut[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[arut/@c]],GCD@@Length/@Split[#]===1&]]/@IntegerPartitions[n-1]];
Table[Length[arut[n]],{n,20}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
MoebiusT(v)={vector(#v, n, sumdiv(n,d,moebius(n/d)*v[d]))}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], MoebiusT(EulerT(v)))); v} \\ Andrew Howroyd, Sep 01 2018

Terms a(21) and beyond from Andrew Howroyd, Sep 01 2018

A317713 Number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 2, 3, 4, 5, 3, 4, 3, 4, 2, 4, 3, 3, 4, 4, 5, 4, 3, 4, 4, 3, 3, 5, 4, 6, 2, 5, 4, 5, 3, 4, 3, 4, 4, 5, 4, 4, 5, 4, 4, 5, 3, 3, 4, 5, 4, 3, 3, 5, 3, 4, 5, 5, 4, 4, 6, 4, 2, 5, 5, 4, 4, 4, 5, 5, 3, 5, 4, 4, 3, 6, 4, 6, 4, 3, 5, 5, 4, 6, 4, 5, 5, 4, 4, 5, 4, 6, 5, 5, 3, 5, 3, 5, 4, 5, 5, 4, 4, 5, 3, 4, 3

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

			20 is the Matula-Goebel number of the tree (oo((o))), which has 4 distinct terminal subtrees: {(oo((o))), ((o)), (o), o}. So a(20) = 4.
See also illustrations in A061773.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Matula-Göbel numbers

One more than A324923.

Cf. A000081, A007097, A049076, A061773, A061775, A076146, A109082, A109129, A206491, A303431, A316476.

ids[n_]:=Union@@FixedPointList[Union@@(Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>PrimePi[p]]&/@#)&,{n}];
Table[Length[ids[n]],{n,100}]

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A324923(n) = { my(lista = List([]), gpf, i); while(n > 1, gpf=A006530(n); i = primepi(gpf); n /= gpf; n *= i; listput(lista,i)); #Set(lista); }; \\ Antti Karttunen, Oct 23 2023
A317713(n) = (1+A324923(n)); \\ Antti Karttunen, Oct 23 2023

a(n) = 1+A324923(n). - Antti Karttunen, Oct 23 2023

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023

A303546 Number of non-isomorphic aperiodic multiset partitions of weight n.

Original entry on oeis.org

1, 3, 9, 29, 90, 285, 909, 2984, 9935, 34113, 119368, 428923, 1574223, 5915235, 22699730, 89000042, 356058539, 1453069854, 6044132793, 25612564200, 110503626702, 485161228675, 2166488899641, 9835209480533, 45370059225227

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime. For this sequence neither the parts nor their multiset union are required to be aperiodic, only the multiset of parts.

			Non-isomorphic representatives of the a(3) = 9 aperiodic multiset partitions are:
  {{1,1,1}}, {{1,2,2}}, {{1,2,3}},
  {{1},{1,1}}, {{1},{2,2}}, {{1},{2,3}}, {{2},{1,2}},
  {{1},{2},{2}}, {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A000740, A000837, A001055, A007716, A007916, A049311, A100953, A255906, A269134, A275024, A293993, A301700, A303386, A303431, A303547.

a(n) = Sum_{d|n} mu(d) * A007716(n/d).

A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 6, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 8, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 3, 2, 1, 8, 2, 2, 2, 4, 1, 8, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1, 5, 1, 4, 5

Offset: 1

Gus Wiseman, May 26 2018

nonn

			The a(60) = 8 factorizations are (2*2*3*5), (2*2*15), (3*4*5), (3*20), (4*15), (5*12), (6*10), (60). Missing from this list are (2*3*10), (2*5*6), (2*30).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001055, A001970, A007716, A034444, A045778, A259936, A281116, A285572, A302242, A303386, A303431, A305001, A305148, A305150.

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],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,100}]

pairwise_indivisible(v) = { for(i=1,#v,for(j=i+1,#v,if(!(v[j]%v[i]),return(0)))); (1); };
A305149(n, m=n, facs=List([])) = if(1==n, pairwise_indivisible(Set(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A305149(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

More terms from Antti Karttunen, Oct 08 2018

A317710 Uniform tree numbers. Matula-Goebel numbers of uniform rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 62, 64, 65, 66, 67, 69, 70, 73, 77, 78, 79, 81, 82, 83, 85, 86, 87, 91, 93, 94, 95, 97

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.

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

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317589.

Cf. A317705, A317707, A317708, A317709, A317711 (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]

A317707 Number of powerful rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 13, 22, 29, 46, 57, 94, 115, 180, 230, 349, 435, 671, 830, 1245, 1572, 2320, 2894, 4287, 5328, 7773, 9752, 14066, 17547, 25328, 31515, 45010, 56289, 79805, 99467, 140778, 175215, 246278, 307273, 429421, 534774, 745776, 927776, 1287038

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

An unlabeled rooted tree is powerful if either it is a single node or a single node with a single powerful tree as a branch, or if the branches of the root all appear with multiplicities greater than 1 and are themselves powerful trees.

			The a(7) = 11 powerful rooted trees:
  ((((((o))))))
  (((((oo)))))
  ((((ooo))))
  ((((o)(o))))
  (((oooo)))
  ((ooooo))
  (((o))((o)))
  ((oo)(oo))
  ((o)(o)(o))
  (oo(o)(o))
  (oooooo)

Alois P. Heinz, Table of n, a(n) for n = 1..8000

Cf. A000081, A001190, A001694, A004111, A301700, A303431, A317102.

Cf. A317705, A317708, A317709, A317710, A317711, A317712, A317718, A317719.

h:= proc(n, k, t) option remember; `if`(k=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, k-j, t+1), j=2..k)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
    end:
a:= proc(n) option remember; `if`(n<2, n, b(n-1$2)+a(n-1)) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 31 2018

purt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Or[Length[#]==1,Min@@Length/@Split[#]>1]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,10}]
(* Second program: *)
h[n_, k_, t_] := h[n, k, t] = If[k == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, k - j, t + 1], {j, 2, k}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* h[a[i], j, 0], {j, 0, n/i}]]];
a[n_] := a[n] = If[n < 2, n, b[n - 1, n - 1] + a[n - 1]];
Array[a, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(27)-a(45) from Alois P. Heinz, Aug 31 2018

A317705 Matula-Goebel numbers of series-reduced powerful rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 196, 256, 343, 361, 392, 512, 784, 1024, 1372, 1444, 1568, 2048, 2401, 2744, 2809, 2888, 3136, 4096, 5488, 5776, 6272, 6859, 8192, 9604, 10976, 11236, 11552, 12544, 16384, 16807, 17161, 17689, 19208, 21952, 22472, 23104, 25088

Offset: 1

Gus Wiseman, Aug 04 2018

nonn

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

			The sequence of Matula-Goebel numbers of series-reduced powerful rooted trees together with the corresponding trees begins:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   32: (ooooo)
   49: ((oo)(oo))
   64: (oooooo)
  128: (ooooooo)
  196: (oo(oo)(oo))
  256: (oooooooo)
  343: ((oo)(oo)(oo))
  361: ((ooo)(ooo))
  392: (ooo(oo)(oo))
  512: (ooooooooo)
  784: (oooo(oo)(oo))

Index entries for sequences related to Matula-Göbel numbers

Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.

Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718, A317719.

powgoQ[n_]:=Or[n==1,And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[1000],powgoQ] (* Gus Wiseman, Aug 31 2018 *)
(* Second program: *)
Nest[Function[a, Append[a, Block[{k = a[[-1]] + 1}, While[Nand[AllTrue[#[[All, -1]], # > 1 & ], AllTrue[PrimePi[#[[All, 1]] ], MemberQ[a, #] &]] &@ FactorInteger@ k, k++]; k]]], {1}, 44] (* Michael De Vlieger, Aug 05 2018 *)

Rewritten by Gus Wiseman, Aug 31 2018

A317708 Number of aperiodic relatively prime trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 20, 48, 108, 255, 595, 1435, 3434, 8372, 20419, 50289, 124289, 309122, 771508, 1934462

Offset: 1

Gus Wiseman, Aug 05 2018

An unlabeled rooted tree is aperiodic and relatively prime iff either it is a single node or a single node with a single aperiodic relatively prime branch, or the branches directly under any given node have empty intersection (relatively prime) and also have relatively prime multiplicities (aperiodic) and are themselves aperiodic relatively prime trees.

			The a(6) = 10 aperiodic relatively prime trees:
  (((((o)))))
  (((o(o))))
  ((o((o))))
  ((oo(o)))
  (o(((o))))
  (o(o(o)))
  ((o)((o)))
  (oo((o)))
  (o(o)(o))
  (ooo(o))

Gus Wiseman, All 48 aperiodic relatively prime trees with 8 nodes.

Cf. A000081, A001190, A004111, A301700, A303431, A316501, A316500.

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

rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,And[Intersection@@#=={},GCD@@Length/@Split[#]==1]]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

cp's OEIS Frontend

A304360 Lexicographically earliest infinite sequence of numbers m > 1 with the property that none of the prime indices of m are in the sequence.

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A303386 Number of aperiodic factorizations of n > 1.

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A301700 Number of aperiodic rooted trees with n nodes.

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A317713 Number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n.

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A303546 Number of non-isomorphic aperiodic multiset partitions of weight n.

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A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

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Mathematica

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A317710 Uniform tree numbers. Matula-Goebel numbers of uniform rooted trees.

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Mathematica

A317707 Number of powerful rooted trees with n nodes.