A294338 -id:A294338 - cp's OEIS frontend

A294336 Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

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

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

Möbius-transform of A294337. - Antti Karttunen, Jun 12 2018

			The a(4096) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.

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

Cf. A000041, A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294337, A294338, A294339, A375598, A375599.

Array[1+Sum[#0[g],{g,Rest[Divisors[GCD@@FactorInteger[#1][[All,2]]]]}]&,200]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d))); \\ Antti Karttunen, Jun 12 2018, after Mathematica-code.

a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} a(d). - Antti Karttunen, Jun 12 2018, after Mathematica-code.

a(n) = A294337(A052409(n)) for n >= 2. - Pontus von Brömssen, Aug 20 2024

More terms from Antti Karttunen, Jun 12 2018

A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 7, 2, 4, 4, 10, 2, 7, 2, 7, 4, 4, 2, 10, 4, 4, 6, 7, 2, 8, 2, 12, 4, 4, 4, 12, 2, 4, 4, 10, 2, 8, 2, 7, 7, 4, 2, 15, 4, 7, 4, 7, 2, 10, 4, 10, 4, 4, 2, 13, 2, 4, 7, 16, 4, 8, 2, 7, 4, 8, 2, 16, 2, 4, 7, 7, 4, 8, 2, 15, 10, 4, 2, 13, 4, 4, 4, 10, 2, 13, 4, 7, 4, 4, 4, 18, 2, 7, 7, 12, 2, 8, 2, 10, 8

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(12) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.

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

Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294338, A294339.

a[n_]:=1+Sum[a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Table[a[2^n],{n,100}]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d)));
A294337(n) = sumdiv(n,d,A294336(d));
\\ Or alternatively, after Mathematica-code as:
A294337(n) = A294336(2^n); \\ Antti Karttunen, Jun 12 2018

a(n) = Sum_{d|n} A294336(d) = A294336(A000079(n)). - Antti Karttunen, Jun 12 2018

More terms from Antti Karttunen, Jun 12 2018

A295931 Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 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, 6, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 29 2017

nonn

By convention a(1) = 1.

Values can be 1, 3, 6, 9, 10, 15, 18, 21, 27, 28, 30, 36, 45, 54, 60, 63, 84, 90, etc. - Robert G. Wilson v, Dec 10 2017

			The a(256) = 10 ways are:
(2^1)^8    (2^2)^4   (2^4)^2  (2^8)^1
(4^1)^4    (4^2)^2   (4^4)^1
(16^1)^2   (16^2)^1
(256^1)^1

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A007425, A052409, A052410, A089723, A093771, A175082, A277562, A281113, A284639, A294786, A294336, A294338, A295920, A295923, A295924, A295935.

f:= proc(n) local m,d,t;
  m:= igcd(seq(t[2],t=ifactors(n)[2]));
  add(numtheory:-tau(d),d=numtheory:-divisors(m))
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Dec 19 2017

Table[Sum[DivisorSigma[0,d],{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

a(A175082(k)) = 1, a(A093771(k)) = 3.

a(n) = Sum_{d|A052409(n)} A000005(d).

A294339 Number of ways to write 2^n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 12, 5, 6, 2, 19, 2, 6, 6, 32, 2, 19, 2, 19, 6, 6, 2, 56, 5, 6, 12, 19, 2, 26, 2, 79, 6, 6, 6, 71, 2, 6, 6, 56, 2, 26, 2, 19, 19, 6, 2, 169, 5, 19, 6, 19, 2, 56, 6, 56, 6, 6, 2, 101, 2, 6, 19, 203, 6, 26, 2, 19, 6, 26, 2, 237, 2, 6, 19, 19

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(6) = 6 ways are 64, 8^2, (2^3)^2, 4^3, (2^2)^3, 2^6.

Hans Havermann, Table of n, a(n) for n = 1..10000

Cf. A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294337, A294338.

f:= proc(n) option remember; local F,t,s,g,a;
  F:= ifactors(n)[2];
  g:= igcd(op(map(t -> t[2],F)));
  t:= 1;
  for s in numtheory:-divisors(g) minus {1} do
    t:= t + procname(mul(a[1]^(a[2]/s),a=F))*procname(s)
  od;
  t
end proc:
seq(f(2^n),n=1..100); # Robert Israel, Dec 01 2017

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

a(n) = A294338(2^n). - R. J. Mathar, Nov 27 2017

A067039 The tower function n^{(n-1)!}.

Original entry on oeis.org

1, 2, 9, 4096, 59604644775390625

Offset: 1

Amarnath Murthy, Dec 29 2001

nonn

a(n) = n^(n-1)^(n-2)^...^3^2^1 with all power operators nested from the left. Nesting from the right gives A049384. - Gus Wiseman, Jul 03 2019

			a(4) = 4^(3!) = 4^6 = 4096.

Cf. A000142, A007489, A049384, A052409, A052410, A140319, A294338, A316782, A325624.

makelist((n+1)^(n!),n,0,6); /* Martin Ettl, Jan 17 2013 */

cp's OEIS Frontend

A294336 Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A295931 Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A294339 Number of ways to write 2^n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A067039 The tower function n^{(n-1)!}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maxima