A279944 -id:A279944

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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.

+10
3

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

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

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..58.

Mathematica Reference, Orderless

EXAMPLE

Triangle begins:

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].

MATHEMATICA

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}]

CROSSREFS

Row lengths are A277996.

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

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Aug 03 2018

STATUS

approved

A279984

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

+10
2

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

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

OFFSET

1,2

LINKS

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

FORMULA

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

MATHEMATICA

nn=100; rads=Select[Range[2, nn], GCD@@FactorInteger[#][[All, 2]]===1&];

Table[Position[rads, Prime[n]][[1, 1]], {n, PrimePi[nn]}]

PROG

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

(Python)

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

CROSSREFS

Cf. A000040, A007916, A279944.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 24 2016

STATUS

approved

A279969

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

+10
1

1, 2, 4, 64, 2085924839766513752338888384931203236916703635113918720651407820138886450957656787131798913024

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..5.

MATHEMATICA

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

CROSSREFS

Cf. A058891, A279944.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 24 2016

STATUS

approved

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.

+10
1

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

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

OFFSET

1,8

LINKS

Table of n, a(n) for n=1..69.

Mathematica Reference, Orderless

EXAMPLE

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 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

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}]

CROSSREFS

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

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

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Aug 03 2018

STATUS

approved

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).

+10
0

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

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

OFFSET

1,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..77.

EXAMPLE

The first nineteen planted achiral trees are:

(o),

((o)), (oo),

(((o))), ((oo)),

((((o)))), (ooo), ((o)(o)), (((oo))),

(((((o))))), ((ooo)), (((o)(o))), ((((oo)))),

((((((o)))))), (oooo), (((ooo))), ((((o)(o)))), (((((oo))))).

MATHEMATICA

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]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 09 2017

STATUS

approved

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