A277615 -id:A277615 - 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

A317658 Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom.

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

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

Given a positive integer n > 1 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)].

Also the number of positions in the orderless Mathematica expression with e-number n.

			The first twenty Mathematica expressions:
   1: o
   2: o[]
   3: o[][]
   4: o[o]
   5: o[][][]
   6: o[o][]
   7: o[][][][]
   8: o[o[]]
   9: o[][o]
  10: o[o][][]
  11: o[][][][][]
  12: o[o[]][]
  13: o[][o][]
  14: o[o][][][]
  15: o[][][][][][]
  16: o[o,o]
  17: o[o[]][][]
  18: o[][o][][]
  19: o[o][][][][]
  20: o[][][][][][][]

Mathematica Reference, Orderless

First differs from A277615 at a(128) = 5, A277615(128) = 6.
Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
Cf. A317652, A317653, A317654, A317655, A317656.

nn=100;
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,x,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[exp[n],{n,1,nn}]

a(rad(x)^(prime(y_1) * ... * prime(y_k))) = a(x) + a(y_1) + ... + a(y_k).
e(2^(2^n)) = o[o,...,o].
e(2^prime(2^prime(2^...))) = o[o[...o[o]]].
e(rad(rad(rad(...)^2)^2)^2) = o[o][o]...[o].

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).

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]

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A007916 Numbers that are not perfect powers.

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A317658 Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom.

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

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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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