A140319 -id:A140319 - cp's OEIS frontend

A167155 Exponential primorial constant Sum_{k>=0} 1/A140319(k).

1, 6, 1, 1, 1, 1, 1, 6, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

M. F. Hasler, Nov 03 2009

This is a Liouville number and therefore transcendental.

			1 + 1/2^1 + 1/3^2 + 1/5^9 + 1/7^(5^9)+ ... = 1.6111116231111111111111111111111111111111...
Since 1/9 = 0.11111... and 1/5^9 = 512*10^(-9), the initial 10 digits are 1.611111623.
Since 1/A140319(4) = 1/7^1953125 = 7.7731519...*10^(-1650583), these digits are followed by a string of 1650573 "1"s, then followed by digits 8884263011....

J. Sondow, Exponential Factorial.
Index entries for transcendental numbers

Cf. A080219.

Clear[ep, s]; ep[0] = 1; ep[n_] := Prime[n]^ep[n-1]; s[n_] := s[n] = RealDigits[Sum[1/ep[k], {k, 0, n}], 10, 105] // First; s[n=1]; While[s[n] != s[n-1], n++]; s[n] (* Jean-François Alcover, Feb 13 2013 *)

1+1/2+1/3^2+1/5^9+1/7^5^9. /* The final dot is part of the code! */

A049384 a(0)=1, a(n+1) = (n+1)^a(n).

Original entry on oeis.org

1, 1, 2, 9, 262144

Offset: 0

Marcel Jackson (Marcel.Jackson(AT)utas.edu.au)

nonn

An "exponential factorial".
Might also be called the "expofactorial" of n. - Walter Arrighetti (walter.arrighetti(AT)fastwebnet.it), Jan 16 2006
By Liouville's theorem, the exponential factorial constant A080219 = Sum_{n>=1} 1/a(n) is a Liouville number and therefore is transcendental. - Jonathan Sondow, Jun 17 2014

			a(4) = 4^9 = 262144.
a(5) = 5^262144 has 183231 decimal digits. - _Rick L. Shepherd_, Feb 15 2002
a(5) = ~6.2060698786608744707483205572846793 * 10^183230. - _Robert G. Wilson v_, Oct 24 2015
a(6) = 6^(5^262144) has 4.829261036048226... * 10^183230 decimal digits. - _Jack Braxton_, Feb 17 2023

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 393-402.
Underwood Dudley, "Mathematical Cranks", MAA 1992, p. 338.
F. Luca, D. Marques, Perfect powers in the summatory function of the power tower, J. Theor. Nombr. Bordeaux 22 (3) (2010) 703, doi:10.5802/jtnb.740

David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.
David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math.NT/0611293, 2006-2007.
Walter Arrighetti, LabCEM, Department of Electronic Engineering, Univ. degli Studi di Roma "La Sapienza".
Walter Arrighetti, Double Vision [Broken link]
Vladimir Orlovsky, Very Big Number, Feb 19 1999
J. Sondow, MathWorld: Exponential Factorial
J. Sondow, Irrationality measures, irrationality bases, and a theorem of Jarnik, arXiv:math/0406300 [math.NT], 2004; see L_4 in Example 4.
Wikipedia, Exponential factorial
Wikipedia, Liouville number

Cf. A000142, A080219, A140319.

Cf. A132859 (essentially the same).

a:= proc(n) option remember;
      `if`(n=0, 1, n^a(n-1))
    end:
seq(a(n), n=0..4);  # Alois P. Heinz, Jan 17 2024

Expofactorial[0] := 1; Expofactorial[n_Integer] := n^Expofactorial[n - 1]; Table[Expofactorial[n], {n, 0, 4}] (* Walter Arrighetti, Jan 24 2006 *)
nxt[{n_,a_}]:={n+1,(n+2)^a}; Transpose[NestList[nxt,{0,1},4]][[2]] (* Harvey P. Dale, May 26 2013 *)

a(n)=if(n>1,n^a(n-1),1) \\ Charles R Greathouse IV, Sep 13 2013

A352492 Powerful numbers whose prime indices are all prime numbers.

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 225, 243, 289, 625, 675, 729, 961, 1089, 1125, 1331, 1681, 2025, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 5625, 6075, 6561, 6889, 7225, 7803, 8649, 9801, 10125, 11881, 11979, 14641, 15125, 15129, 15625, 16129, 16875

Offset: 1

Gus Wiseman, Mar 24 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices (not prime factors) begin:
    1: {}
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   81: {2,2,2,2}
  121: {5,5}
  125: {3,3,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  289: {7,7}
  625: {3,3,3,3}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  961: {11,11}
For example, 675 = prime(2)^3 prime(3)^2 = 3^3 * 5^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Powerful numbers are A001694, counted by A007690.
The version for prime exponents instead of indices is A056166, counted by A055923.
This is the powerful case of A076610 (products of A006450), counted by A000607.
The partitions with these Heinz numbers are counted by A339218.
A000040 lists primes.
A031368 lists primes of odd index, products A066208.
A101436 counts exponents in prime factorization that are themselves prime.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A053810 lists all numbers p^q with p and q prime, counted by A230595.
A257994 counts prime indices that are themselves prime, complement A330944.
Cf. A000720, A000961, A001597, A007821, A007916, A052485, A109297, A140319, A164336, A181819, A325131, A330945, A346068.

Select[Range[1000],#==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]

Intersection of A001694 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/(p*(p-1))) = 1.24410463... - Amiram Eldar, May 04 2022

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 */

A260548 a(n) = prime(1)^prime(2)^prime(3)^...^prime(n).

Original entry on oeis.org

2, 8, 14134776518227074636666380005943348126619871175004951664972849610340958208

Offset: 1

Peter Woodward, Jul 29 2015

nonn

			a(1) = 2; a(2) = 2^3; a(3) = 2^3^5.

Cf. A140319, A248012.

A152859 Tower of prime powers: a(n)=prime(n)^a(n-1), a(0)=0.

Original entry on oeis.org

0, 1, 3, 125, 4337654948097993282537354757263188251697832994620405101744893017744569432720994168089672192211758909320807

Offset: 0

ShaoJun Ying (dolphinysj(AT)gmail.com), Dec 14 2008

nonn

Originally called "Exprimorial numbers (exponential prime factorials)", the strict analog would be "exponential primorial". [Editor's Note]

			a(4) = 7 ^ a(3) = 7 ^ 125.
a(5) = 11 ^ a(4) has approximately 4.5 * 10^105 digits, starting with 335856... and ending in ...815171.

J. Sondow, Exponential Factorial
Wikipedia, Factorial

Cf. A002110, A049384.

Cf. A140319 (alternate definition: start with a(0)=1). - Paolo P. Lava, Jul 31 2018

unsigned long Exprimorial(unsigned int n) {
if (n == 0) return 0;
return pow(prime(n),Exprimorial(n - 1));
}

vector(4,i,t=if(i==1,1,prime(i)^t)) /* indices are shifted by 1 */ \\ M. F. Hasler, Nov 01 2009

a(n) = 0 if n = 0; a(n) = prime(n) ^ a(n - 1), n >= 1.

Edited by M. F. Hasler, Nov 01 2009

cp's OEIS Frontend

A167155 Exponential primorial constant Sum_{k>=0} 1/A140319(k).

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A049384 a(0)=1, a(n+1) = (n+1)^a(n).

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A352492 Powerful numbers whose prime indices are all prime numbers.

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

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A260548 a(n) = prime(1)^prime(2)^prime(3)^...^prime(n).

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A152859 Tower of prime powers: a(n)=prime(n)^a(n-1), a(0)=0.

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