A279944 -id:A279944 - cp's OEIS frontend

A317676 Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

1, 2, 3, 4, 5, 6, 8, 9, 16, 7, 10, 12, 13, 21, 25, 27, 32, 36, 64, 81, 128, 256, 11, 14, 17, 18, 28, 33, 35, 41, 45, 49, 75, 93, 100, 125, 144, 145, 169, 216, 243, 279, 441, 512, 625, 729, 1024, 1296, 2048, 2187, 4096, 6561, 8192, 16384, 65536, 524288, 8388608, 9007199254740992

Offset: 1

Gus Wiseman, Aug 03 2018

Given a positive integer n we construct a unique free pure symmetric multifunction e(n) by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)].

Every free pure symmetric multifunction (with empty expressions allowed) f with one atom and n positions has a unique e-number n such that e(n) = f, and vice versa, so this sequence is a permutation of the positive integers.

			Triangle begins:
  1
  2
  3   4
  5   6   8   9  16
  7  10  12  13  21  25  27  32  36  64  81 128 256
Corresponding triangle of free pure symmetric multifunctions (with empty expressions allowed) begins:
  o,
  o[],
  o[][], o[o],
  o[][][], o[o][], o[o[]], o[][o], o[o,o].

Mathematica Reference, Orderless

Row lengths are A277996.

Cf. A007916, A052409, A052410, A052893, A053492, A215366, A279944, A280000, A299759, A317658, A317659, A317677.

maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];
radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]==1];
Clear[rad];rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
ungo[x_?AtomQ]:=1;ungo[h_[g___]]:=rad[ungo[h]]^(Times@@Prime/@ungo/@{g});
Table[Sort[ungo/@maxUsing[n]],{n,5}]

A279984 Positions of the prime numbers in the sequence of numbers that are not perfect powers (A007916).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 14, 18, 22, 24, 28, 32, 34, 38, 43, 49, 51, 56, 60, 62, 68, 71, 77, 85, 88, 90, 94, 96, 100, 112, 115, 121, 123, 132, 134, 140, 146, 150, 155, 161, 163, 173, 175, 178, 180, 192, 203, 206, 208, 212, 218, 220, 229, 234, 240, 246, 248, 254

Offset: 1

Gus Wiseman, Dec 24 2016

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000040, A007916, A279944.

nn=100;rads=Select[Range[2,nn],GCD@@FactorInteger[#][[All,2]]===1&];
Table[Position[rads,Prime[n]][[1,1]],{n,PrimePi[nn]}]

lista(nn) = Vec(select(x->isprime(x), Vec(select(x->(!ispower(x)&&x>1), [1..nn])), 1)); \\ Michel Marcus, May 04 2018

from sympy import prime, mobius, integer_nthroot
def A279984(n): return int((p:=prime(n))-1+sum(mobius(k)*(integer_nthroot(p,k)[0]-1) for k in range(2,p.bit_length()))) # Chai Wah Wu, Oct 12 2024

A007916(a(n)) = A000040(n).

A279969 a(1)=1, a(n+1)=2^(prime(a(n))-1).

Original entry on oeis.org

1, 2, 4, 64, 2085924839766513752338888384931203236916703635113918720651407820138886450957656787131798913024

Offset: 1

Gus Wiseman, Dec 24 2016

nonn

Cf. A058891, A279944.

Mathematica
```
NestList[Power[2,Prime[#]-1]&,1,4]
```

A317659 Regular triangle where T(n,k) is the number of distinct free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 43, 33, 9, 1, 0, 1, 21, 92, 118, 55, 11, 1, 0, 1, 28, 174, 341, 252, 82, 13, 1, 0, 1, 36, 302, 845, 935, 463, 115, 15, 1, 0, 1, 45, 490, 1864, 2921, 2103, 769, 153, 17, 1, 0, 1, 55, 755

Offset: 1

Gus Wiseman, Aug 03 2018

			The T(5,3) = 5 expressions are o[o[o]], o[o,o[]], o[][o,o], o[o][o], o[o,o][].
Triangle begins:
    1
    1    0
    1    1    0
    1    3    1    0
    1    6    5    1    0
    1   10   17    7    1    0
    1   15   43   33    9    1    0
    1   21   92  118   55   11    1    0
    1   28  174  341  252   82   13    1    0
    1   36  302  845  935  463  115   15    1    0
    1   45  490 1864 2921 2103  769  153   17    1    0
    1   55  755 3755 7981 8012 4145 1187  197   19    1    0

Mathematica Reference, Orderless

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317655, A317656, A317676.

maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];
Table[Length[Select[maxUsing[n],Length[Position[#,"o"]]==k&]],{n,12},{k,n}]

A292127 a(1) = 1, a(r(n)^k) = 1 + k * a(n) where r(n) is the n-th number that is not a perfect power A007916(n).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 7, 7, 8, 6, 7, 8, 8, 7, 9, 7, 7, 8, 9, 9, 6, 8, 10, 8, 7, 8, 9, 10, 10, 7, 9, 11, 9, 8, 9, 10, 11, 9, 11, 8, 10, 12, 10, 9, 10, 11, 12, 10, 12, 9, 11, 13, 7, 11, 10, 11, 12, 13, 11, 13, 10, 12, 14, 8, 12, 11

Offset: 1

Gus Wiseman, Sep 09 2017

nonn

Any positive integer greater than 1 can be written uniquely as a perfect power r(n)^k. We define a planted achiral (or generalized Bethe) tree b(n) for any positive integer greater than 1 by writing n as a perfect power r(d)^k and forming a tree with k branches all equal to b(d). Then a(n) is the number of nodes in b(n).

			The first nineteen planted achiral trees are:
o,
(o),
((o)), (oo),
(((o))), ((oo)),
((((o)))), (ooo), ((o)(o)), (((oo))),
(((((o))))), ((ooo)), (((o)(o))), ((((oo)))),
((((((o)))))), (oooo), (((ooo))), ((((o)(o)))), (((((oo))))).

Cf. A003238, A007916, A052409, A052410, A061775, A214577, A277576, A277615, A278028, A279614, A279944, A289023.

nn=100;
rads=Select[Range[2,nn],GCD@@FactorInteger[#][[All,2]]===1&];
a[1]:=1;a[n_]:=With[{k=GCD@@FactorInteger[n][[All,2]]},1+k*a[Position[rads,n^(1/k)][[1,1]]]];
Array[a,nn]

cp's OEIS Frontend

A317676 Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

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A279984 Positions of the prime numbers in the sequence of numbers that are not perfect powers (A007916).

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A279969 a(1)=1, a(n+1)=2^(prime(a(n))-1).

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A317659 Regular triangle where T(n,k) is the number of distinct free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and k leaves.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A292127 a(1) = 1, a(r(n)^k) = 1 + k * a(n) where r(n) is the n-th number that is not a perfect power A007916(n).

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Mathematica