A294336 -id:A294336 - cp's OEIS frontend

A304481 Turn the power-tower for n upside-down.

1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 32, 26, 27, 28, 29, 30, 31, 25, 33, 34, 35, 64, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 128, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 36, 65, 66, 67

Offset: 1

Gus Wiseman, May 13 2018

This is an involution of the positive integers.

The power-tower for n is defined as follows. Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. Then a(n) = c(x_k)^...^c(x_3)^c(x_2)^c(x_1).

			The power tower of 81 is 3^2^2, which turned upside-down is 2^2^3 = 256, so a(81) = 256.

Robert Israel, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304491, A304492, A304495.

f:= proc(n,r) local F,a,y;
     if n = 1 then return 1 fi;
     F:= ifactors(n)[2];
     y:= igcd(seq(t[2],t=F));
     if y = 1 then return n^r fi;
     a:= mul(t[1]^(t[2]/y),t=F);
     procname(y,a^r)
end proc:
seq(f(n,1),n=1..100); # Robert Israel, May 13 2018

tow[n_]:=If[n==1,{},With[{g=GCD@@FactorInteger[n][[All,2]]},If[g===1,{n},Prepend[tow[g],n^(1/g)]]]];
Table[Power@@Reverse[tow[n]],{n,100}]

A304491 Last or deepest exponent in the power-tower for n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 3, 2, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 2, 26, 3, 28, 29, 30, 31, 5, 33, 34, 35, 2, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 2, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 6, 65, 66, 67, 68, 69

Offset: 1

Gus Wiseman, May 13 2018

nonn

Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. Then a(n) = c(x_k).

			We have 16 = 2^2^2, so a(16) = 2.
We have 64 = 2^6, so a(64) = 6.
We have 81 = 3^2^2, so a(81) = 2.
We have 256 = 2^2^3, so a(256) = 3.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304481, A304492, A304495.

a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},If[g==1,n,a[g]]]];
Array[a,100]

a(n)={my(t=n); while(t, n=t; t=ispower(t)); n} \\ Andrew Howroyd, Aug 26 2018

a(n) = A007916(A278028(n, A288636(n))).

A304495 Decapitate the power-tower for n, i.e., remove the last (deepest) exponent.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 13 2018

nonn

a(1) = 0 by convention.

Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. Then a(n) = c(x_1)^c(x_2)^c(x_3)^...^c(x_{k-1}).

			We have 64 = 2^6, so a(64) = 2.
We have 216 = 6^3, so a(216) = 6.
We have 256 = 2^2^3, so a(256) = 2^2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A052409, A052410, A001597, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304481, A304491, A304492.

tow[n_]:=If[n==1,{},With[{g=GCD@@FactorInteger[n][[All,2]]},If[g===1,{n},Prepend[tow[g],n^(1/g)]]]];
Table[If[n==1,0,Power@@Most[tow[n]]],{n,100}]

A304495(n) = if(1==n,0,my(e, r, tow = List([])); while((e = ispower(n,,&r)) > 1, listput(tow, r); n = e;); n = 1; while(length(tow)>0, e = tow[#tow]; listpop(tow); n = e^n;); (n)); \\ Antti Karttunen, Jul 23 2018

a(m) <> 1 if m is a perfect power (A001597). - Michel Marcus, Jul 23 2018

Name edited and more terms from Antti Karttunen, Jul 23 2018

A317100 Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 3, 5, 12, 17, 41, 65, 144, 262, 533, 1025, 2110, 4097, 8261, 16407, 32928, 65537, 131384, 262145, 524854, 1048647, 2098181, 4194305, 8390924, 16777234, 33558533, 67109132, 134226070, 268435457, 536887919, 1073741825, 2147516736, 4294968327, 8590000133

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

In these trees, achiral means that all branches directly under any given node that is not a leaf or a cover of leaves are equal, and series-reduced means that every node that is not a leaf or a cover of leaves has at least two branches.

			The a(4) = 12 trees:
  (1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)),
  (1222),
  (1122), ((12)(12)),
  (1112),
  (1233),
  (1223),
  (1123),
  (1234).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001678, A003238, A052409, A052410, A067824, A167865, A168532, A214577, A289078, A294336, A316782, A317099.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
b[n_]:=1+Sum[b[n/d],{d,Rest[Divisors[n]]}];
a[n_]:=Sum[b[GCD@@Length/@Split[ptn]],{ptn,allnorm[n]}];
Array[a,10]

seq(n)={my(v=vector(n)); for(n=1, n, v[n]=2^(n-1) + sumdiv(n, d, v[d])); v} \\ Andrew Howroyd, Aug 19 2018

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 07 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018

A304492 Position in the sequence of numbers that are not perfect powers (A007916) of the last or deepest exponent in the power-tower for n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 3, 2, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 16, 17, 18, 19, 20, 2, 21, 3, 22, 23, 24, 25, 4, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 2, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 5, 55, 56, 57, 58, 59, 60

Offset: 1

Gus Wiseman, May 13 2018

nonn

Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. Then a(n) = x_k.

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304481, A304491, A304495.

nn=100;
a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},If[g==1,n,a[g]]]];
rads=Union[Array[a,nn]];
Table[a[n],{n,nn}]/.Table[rads[[i]]->i,{i,Length[rads]}]

a(n) = A278028(n, A288636(n)).

A316789 Number of same-tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

A constant factorization of n is a finite nonempty constant multiset of positive integers greater than 1 with product n. Constant factorizations correspond to perfect divisors (A089723). A same-tree-factorization of n is either (case 1) the number n itself or (case 2) a finite sequence of two or more same-tree-factorizations, one of each factor in a constant factorization of n.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(64) = 14 same-tree-factorizations:
  64
  (8*8)
  (4*4*4)
  (8*(2*2*2))
  ((2*2*2)*8)
  (4*4*(2*2))
  (4*(2*2)*4)
  ((2*2)*4*4)
  (2*2*2*2*2*2)
  (4*(2*2)*(2*2))
  ((2*2)*4*(2*2))
  ((2*2)*(2*2)*4)
  ((2*2*2)*(2*2*2))
  ((2*2)*(2*2)*(2*2))

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A001055, A001597, A001678, A003238, A007916, A052409, A052410, A067824, A089723, A281118, A281145, A294336, A316790.

a[n_]:=1+Sum[a[n^(1/d)]^d,{d,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}]
Array[a,100]

a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(d>1, a(z^(e/d))^d)))} \\ Andrew Howroyd, Nov 18 2018

a(n) = 1 + Sum_{n = x^y, y > 1} a(x)^y.

a(2^n) = A281145(n).

A375379 Value of the power tower formed by the numbers obtained by adding 1 to the terms in the n-th composition (in standard order).

Original entry on oeis.org

1, 2, 3, 4, 4, 9, 8, 16, 5, 16, 27, 81, 16, 512, 256, 65536, 6, 25, 64, 256, 81, 19683, 6561, 43046721, 32, 65536, 134217728, 2417851639229258349412352, 65536

Offset: 0

Pontus von Brömssen, Aug 14 2024

nonn

This is a version of A375378 that does not allow 1's in the power towers.
It is natural to define a(0) = 1.
The number k >= 1 appears A294336(k) times as a term in this sequence.

			For n = 31, the 31st composition is (1, 1, 1, 1, 1), so a(31) = 2^2^2^2^2 = 2^65536.
For n = 37, the 37th composition is (3,2,1), so a(37) = 4^3^2 = 4^9 = 262144.

Pontus von Brömssen, Table of n, a(n) for n = 0..30

Cf. A053645, A065120, A066099 (compositions in standard order), A294336, A375378.

a(n) = (A065120(n)+1)^a(A053645(n)) for n >= 1.

cp's OEIS Frontend

A304481 Turn the power-tower for n upside-down.

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A304491 Last or deepest exponent in the power-tower for n.

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A304495 Decapitate the power-tower for n, i.e., remove the last (deepest) exponent.

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A317100 Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.

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A304492 Position in the sequence of numbers that are not perfect powers (A007916) of the last or deepest exponent in the power-tower for n.

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Mathematica

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A316789 Number of same-tree-factorizations of n.

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Mathematica

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A375379 Value of the power tower formed by the numbers obtained by adding 1 to the terms in the n-th composition (in standard order).

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