A277576 -id:A277576 - cp's OEIS frontend

A007916 Numbers that are not perfect powers.

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

R. Muller

From Gus Wiseman, Oct 23 2016: (Start)
There is a 1-to-1 correspondence between integers N >= 2 and sequences a(x_1),a(x_2),...,a(x_k) of terms from this sequence. Every N >= 2 can be written uniquely as a "power tower"
N = a(x_1)^a(x_2)^a(x_3)^...^a(x_k),
where the exponents are to be nested from the right.
Proof: If N is not a perfect power then N = a(x) for some x, and we are done. Otherwise, write N = a(x_1)^M for some M >=2, and repeat the process. QED
Of course, prime numbers also have distinct power towers (see A164336). (End)
These numbers can be computed with a modified Sieve of Eratosthenes: (1) start at n=2; (2) if n is not crossed out, then append n to the sequence and cross out all powers of n; (3) set n = n+1 and go to step 2. - Sam Alexander, Dec 15 2003
These are all numbers such that the multiplicities of the prime factors have no common divisor. The first number in the sequence whose prime multiplicities are not coprime is 180 = 2 * 2 * 3 * 3 * 5. Mathematica: CoprimeQ[2,2,1]->False. - Gus Wiseman, Jan 14 2017

			Example of the power tower factorizations for the first nine positive integers: 1=1, 2=a(1), 3=a(2), 4=a(1)^a(1), 5=a(3), 6=a(4), 7=a(5), 8=a(1)^a(2), 9=a(2)^a(1). - _Gus Wiseman_, Oct 20 2016

N. J. A. Sloane, Table of n, a(n) for n = 1..9875
Joakim Munkhammar, The Riemann zeta function as a sum of geometric series, The Mathematical Gazette (2020) Vol. 104, Issue 561, 527-530.
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993
Index entries for sequences generated by sieves

Complement of A001597. Union of A052485 and A052486.
Cf. A144338, A277562, A277564, A075802.
Cf. A153158 (squares of these numbers).
See A277562, A277564, A277576, A277615 for more about the power towers.
A278029 is a left inverse.
Cf. A052409.

a007916 n = a007916_list !! (n-1)
a007916_list = filter ((== 1) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

[n : n in [2..1000] | not IsPower(n) ];

Maple
```
See link.
```

a = {}; Do[If[Apply[GCD, Transpose[FactorInteger[n]][[2]]] == 1, a = Append[a, n]], {n, 2, 200}];
Select[Range[2,200],GCD@@FactorInteger[#][[All,-1]]===1&] (* Michael De Vlieger, Oct 21 2016. Corrected by Gus Wiseman, Jan 14 2017 *)

is(n)=!ispower(n)&&n>1 \\ Charles R Greathouse IV, Jul 01 2013

from sympy import mobius, integer_nthroot
def A007916(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 13 2024

A075802(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2009
Gcd(exponents in prime factorization of a(n)) = 1, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
a(n) ~ n. - Charles R Greathouse IV, Jul 01 2013
A052409(a(n)) = 1. - Ridouane Oudra, Nov 23 2024

More terms from Henry Bottomley, Sep 12 2000
Edited by Charles R Greathouse IV, Mar 18 2010
Further edited by N. J. A. Sloane, Nov 09 2016

A279944 Number of positions in the free pure symmetric multifunction in one symbol with j-number n.

Original entry on oeis.org

1, 3, 5, 5, 7, 7, 9, 4, 7, 9, 11, 6, 9, 11, 13, 7, 8, 11, 13, 15, 9, 10, 13, 15, 9, 17, 6, 11, 12, 15, 17, 6, 11, 19, 8, 9, 13, 14, 17, 19, 8, 13, 21, 10, 11, 15, 16, 19, 11, 21, 10, 15, 23, 12, 13, 17, 18, 21, 13, 23, 12, 17, 25, 7, 14, 15, 19, 20, 23, 15, 25, 14, 19, 27, 9, 16, 17, 21, 22, 25, 9, 17, 27, 16, 21, 29, 11, 18, 19, 23, 24, 27, 11, 19, 29, 18, 23, 31, 13, 11

Offset: 1

Gus Wiseman, Dec 24 2016

nonn

A free pure symmetric multifunction in one symbol f in PSM(x) is either (case 1) f = the symbol x, or (case 2) f = an expression of the form h[g_1,...,g_k] where h is in PSM(x), each of the g_i for i=1..(k>0) is in PSM(x), and for i < j we have g_i <= g_j under a canonical total ordering of PSM(x), such as the Mathematica ordering of expressions. For a positive integer n we define a free pure symmetric multifunction j(n) by: j(1)=x; j(n>1) = j(h)[j(g_1),...,j(g_k)] where n = r(h)^(p(g_1)*...*p(g_k)-1). Here r(n) is the n-th number that is not a perfect power (A007916) and p(n) is the n-th prime number (A000040). See example. Then a(n) is the number of brackets [...] plus the number of x's in j(n).

			The first 20 free pure symmetric multifunctions in x are:
j(1)  = j(1)            = x
j(2)  = j(1)[j(1)]      = x[x]
j(3)  = j(2)[j(1)]      = x[x][x]
j(4)  = j(1)[j(2)]      = x[x[x]]
j(5)  = j(3)[j(1)]      = x[x][x][x]
j(6)  = j(4)[j(1)]      = x[x[x]][x]
j(7)  = j(5)[j(1)]      = x[x][x][x][x]
j(8)  = j(1)[j(1),j(1)] = x[x,x]
j(9)  = j(2)[j(2)]      = x[x][x[x]]
j(10) = j(6)[j(1)]      = x[x[x]][x][x]
j(11) = j(7)[j(1)]      = x[x][x][x][x][x]
j(12) = j(8)[j(1)]      = x[x,x][x]
j(13) = j(9)[j(1)]      = x[x][x[x]][x]
j(14) = j(10)[j(1)]     = x[x[x]][x][x][x]
j(15) = j(11)[j(1)]     = x[x][x][x][x][x][x]
j(16) = j(1)[j(3)]      = x[x[x][x]]
j(17) = j(12)[j(1)]     = x[x,x][x][x]
j(18) = j(13)[j(1)]     = x[x][x[x]][x][x]
j(19) = j(14)[j(1)]     = x[x[x]][x][x][x][x]
j(20) = j(15)[j(1)]     = x[x][x][x][x][x][x][x].

Cf. A005043, A007916, A106490, A277564, A277615, A277996, A278028, A280000.

Cf. A279984 (numbers j(n)[x]=j(prime(n))), A277576 (numbers j(n)=x[x][x][x]...), A058891 (numbers j(n)=x[x,...,x]), A279969 (numbers j(n)=x[x[...[x]]]).

nn=100;
radQ[n_]:=If[n===1,False,SameQ[GCD@@FactorInteger[n][[All,2]],1]];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Set@@@Array[radPi[rad[#]]==#&,nn];
jfac[n_]:=With[{g=GCD@@FactorInteger[n+1][[All,2]]},JIX[radPi[Power[n+1,1/g]],Flatten[Cases[FactorInteger[g+1],{p_,k_}:>ConstantArray[PrimePi[p],k]]]]];
diwt[n_]:=If[n===1,1,Apply[1+diwt[#1]+Total[diwt/@#2]&,jfac[n-1]]];
Array[diwt,nn]

a(A007916(h)^(A000040(g_1)*...*A000040(g_k)-1)) = 1 + a(h) + a(g_1) + ... + a(g_k).

A316112 Number of leaves in the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 3, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 1, 3

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction e(n) with one atom 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)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			e(21025) = o[o[o]][o] has 4 leaves so a(21025) = 4.

Cf. A007916, A052409, A052410, A109129, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.

nn=1000;
radQ[n_]:=If[n==1,False,GCD@@FactorInteger[n][[All,2]]==1];
rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},a[radPi[Power[n,1/g]]]+Sum[a[PrimePi[pr[[1]]]]*pr[[2]],{pr,If[g==1,{},FactorInteger[g]]}]]];
Table[a[n],{n,100}]

a(rad(x)^(prime(y_1) * ... * prime(y_k))) = a(x) + a(y_1) + ... + a(y_k) where rad = A007916.

A317056 Depth of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 3, 4, 6, 1, 4, 4, 5, 7, 2, 5, 5, 6, 3, 8, 2, 3, 6, 6, 7, 3, 4, 9, 3, 2, 4, 7, 7, 8, 4, 5, 10, 4, 3, 5, 8, 8, 4, 9, 5, 6, 11, 5, 4, 6, 9, 9, 5, 10, 6, 7, 12, 2, 6, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 7, 6, 8, 11, 11, 2, 7, 12

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction e(n) with one atom 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)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			e(21025) = o[o[o]][o] has depth 3 so a(21025) = 3.

Cf. A007916, A052409, A052410, A109082, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.

nn=1000;
radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n===1,"o",With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Table[Max@@Length/@Position[exp[n],_],{n,200}]

A317994 Number of inequivalent leaf-colorings of the free pure symmetric multifunction with e-number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 1, 4, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 5, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 5, 2, 5, 4, 2, 2, 4, 2, 2, 2, 2, 1, 5

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom 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)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			Inequivalent representatives of the a(441) = 11 colorings of the expression e(441) = o[o,o][o] are the following.
  1[1,1][1]
  1[1,1][2]
  1[1,2][1]
  1[1,2][2]
  1[1,2][3]
  1[2,2][1]
  1[2,2][2]
  1[2,2][3]
  1[2,3][1]
  1[2,3][2]
  1[2,3][4]

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317765.

A317765 Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom 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)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			The a(12) = 4 subexpressions of o[o[]][] are {o, o[], o[o[]], o[o[]][]}.

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317713, A317994.

nn=1000;
radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n===1,"o",With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Table[Length[Union[Cases[exp[n],_,{0,Infinity},Heads->True]]],{n,100}]

A277615 a(1)=1; thereafter, if n = c(x_1)^...^c(x_k) (where c(k) = A007916(k) and with parentheses nested from the right, as in the definition of A277564), a(n) = 1 + a(x_1) + ... + a(x_k).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 4, 5, 6, 5, 5, 6, 7, 4, 6, 6, 7, 8, 5, 7, 7, 8, 5, 9, 5, 6, 8, 8, 9, 5, 6, 10, 6, 5, 7, 9, 9, 10, 6, 7, 11, 7, 6, 8, 10, 10, 6, 11, 7, 8, 12, 8, 7, 9, 11, 11, 7, 12, 8, 9, 13, 5, 9, 8, 10, 12, 12, 8, 13, 9, 10, 14, 6, 10, 9, 11, 13, 13, 5, 9, 14, 10, 11, 15, 7, 11, 10, 12, 14, 14, 6, 10, 15, 11, 12

Offset: 1

Gus Wiseman, Oct 23 2016

A007916 lists the numbers whose prime multiplicities are relatively prime. For each n we can construct a plane tree by repeatedly factoring all positive integers at any level into their corresponding power towers of non-perfect-powers (see A277564). a(n) is the number of nodes in this plane tree.

			a(1)=1, a(2)=1+a(1)=2, a(3)=1+a(2)=3, a(4)=1+a(1)+a(1)=3 because 4=c(1)^c(1), a(8)=1+a(1)+a(2)=4 because 8=c(1)^c(2), a(9)=1+a(2)+a(1)=4 because 9=c(2)^c(1), a(10)=1+a(6)=5 because 10=c(6).

Gus Wiseman, Table of n, a(n) for n = 1..10000

Cf. A000108, A007916, A014221, A277564, A277576.

nn=10000;
radicalQ[1]:=False;radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
hyperfactor[1]:={};hyperfactor[n_?radicalQ]:={n};
hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
rad[0]:=1;rad[n_?Positive]:=rad[n]=NestWhile[#+1&,rad[n-1]+1,Not[radicalQ[#]]&];Set@@@Array[radPi[rad[#]]==#&,nn];
rnk[n_]:=rnk[n]=1+Total[rnk/@radPi/@hyperfactor[n]];
Array[rnk,nn]

First appearance of n is a(A277576(n)). Last appearance of n is a(2^^{n-1}) where ^^ denotes iterated exponentiation (or tetration).

Number of appearances of n is the Catalan number |{k:a(k)=n}| = C_{n-1}.

Edited by N. J. A. Sloane, Nov 09 2016

A318149 e-numbers of free pure symmetric multifunctions with one atom.

Original entry on oeis.org

1, 4, 16, 36, 128, 256, 441, 1296, 2025, 16384, 21025, 65536, 77841, 194481, 220900, 279936, 1679616, 1803649, 4100625, 4338889, 268435456, 273571600, 442050625, 449482401, 1801088541, 4294967296, 4334247225, 6059221281

Offset: 1

Gus Wiseman, Aug 19 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique orderless expression e(n) (as can be represented in functional programming languages such as Mathematica) with one atom 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)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1). The sequence consists of all numbers n such that e(n) contains no empty subexpressions f[].

			The sequence of free pure symmetric multifunctions with one atom "o", together with their e-numbers begins:
       1: o
       4: o[o]
      16: o[o,o]
      36: o[o][o]
     128: o[o[o]]
     256: o[o,o,o]
     441: o[o,o][o]
    1296: o[o][o,o]
    2025: o[o][o][o]
   16384: o[o,o[o]]
   21025: o[o[o]][o]
   65536: o[o,o,o,o]
   77841: o[o,o,o][o]
  194481: o[o,o][o,o]
  220900: o[o,o][o][o]
  279936: o[o][o[o]]

Charlie Neder, Table of n, a(n) for n = 1..310
Charlie Neder, Python program for calculating this sequence

A subsequence of A001597.
Cf. A007916, A052409, A052410, A277576, A277996, A280000.
Cf. A317658, A316112, A317056, A317765, A317994, A318150, A318152, A318153.

nn=1000;
radQ[n_]:=If[n==1,False,GCD@@FactorInteger[n][[All,2]]==1];
rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n==1,"o",With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Select[Range[nn],FreeQ[exp[#],_[]]&]

Python
```
See Neder link.
```

a(16)-a(27) from Charlie Neder, Sep 01 2018

A318150 e-numbers of free pure functions with one atom.

Original entry on oeis.org

1, 4, 36, 128, 2025, 21025, 279936, 4338889, 449482401, 78701569444, 373669453125, 18845583322500, 1347646586640625, 202054211912421649, 6193981883008128893161, 139629322539586311507076, 170147232533595290155627, 355156175404848064835984400

Offset: 1

Gus Wiseman, Aug 19 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique orderless expression e(n) (as can be represented in functional programming languages such as Mathematica) with one atom 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)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1). This sequence consists of all numbers n such that e(n) contains no non-unitary subexpressions f[x_1, ..., x_k] where k != 1.

			The sequence of all free pure functions with one atom together with their e-numbers begins:
        1: o
        4: o[o]
       36: o[o][o]
      128: o[o[o]]
     2025: o[o][o][o]
    21025: o[o[o]][o]
   279936: o[o][o[o]]
  4338889: o[o][o][o][o]

Charlie Neder, Table of n, a(n) for n = 1..44

A subsequence of A001597.
Cf. A000108, A007916, A052409, A052410, A277576, A277996, A280000.
Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318152, A318153.

a(1) = 1, and if a and b are in this sequence then so is rad(a)^prime(b). - Charlie Neder, Feb 23 2019

More terms from Charlie Neder, Feb 23 2019

A318152 e-numbers of unlabeled rooted trees. A number n is in the sequence iff n = 2^(prime(y_1) * ... * prime(y_k)) for some k > 0 and y_1, ..., y_k already in the sequence.

Original entry on oeis.org

1, 4, 16, 128, 256, 16384, 65536, 268435456, 4294967296, 562949953421312, 9007199254740992, 72057594037927936, 18446744073709551616, 316912650057057350374175801344, 81129638414606681695789005144064, 5192296858534827628530496329220096

Offset: 1

Gus Wiseman, Aug 19 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique orderless expression e(n) (as can be represented in functional programming languages such as Mathematica) with one atom 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)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1). The sequence consists of all numbers n such that e(n) contains no empty subexpressions f[] or subexpressions in heads f[x_1, ..., x_k][y_1, ..., y_k] where k,j >= 0.

			The sequence contains 16384 = 2^14 = 2^(prime(1) * prime(4)) because 1 and 4 both already belong to the sequence.
The sequence of unlabeled rooted trees with e-numbers in the sequence begins:
      1: o
      4: (o)
     16: (oo)
    128: ((o))
    256: (ooo)
  16384: (o(o))
  65536: (oooo)
    .    (oo(o))
    .    (ooooo)
    .    ((o)(o))
         ((oo))
         (ooo(o))
         (oooooo)
         (o(o)(o))
         (o(oo))
         (oooo(o))
         (ooooooo)
         (oo(o)(o))

A subsequence of A000079 and A318151.
Cf. A000081, A007916, A052409, A052410, A277576, A277996, A280000.
Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318153.

baQ[n_]:=Or[n==1,MatchQ[FactorInteger[n],{{2,?(And@@Cases[FactorInteger[#],{p,k_}:>baQ[PrimePi[p]]]&)}}]];
Select[2^Range[0,50],baQ]

cp's OEIS Frontend

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A316112 Number of leaves in the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

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A318149 e-numbers of free pure symmetric multifunctions with one atom.

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