A167155 -id:A167155 - cp's OEIS frontend

A140319 Exponential primorial: a(n)=prime(n)^a(n-1), a(0)=1, where prime(n)=A000040(n).

1, 2, 9, 1953125

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 26 2008

The next term is too large to include.

			a(1) = prime(1)^a(0) = 2^1 = 2.
a(2) = 3^2 = 9.
a(3) = 5^9 = 1953125.
a(4) = 7^1953125 has 1650583 digits, starting with 12864794... and ending in ...31920807. - _M. F. Hasler_, Nov 03 2009

Jonathan Sondow, Exponential Factorial, Eric Weisstein's World of Mathematics.

Cf. A002110, A049384, A139802, A167155.

Cf. A152859 (alternate definition: start with a(0)=0), A167155.

P:=proc(n) local a,i; a:=2; print(a); for i from 2 by 1 to n do a:=ithprime(i)^a; print(a); od; end: P(5);

a[0]=1;a[n_]:=Prime[n]^a[n-1];Array[a,4,0] (* James C. McMahon, Jul 08 2025 *)

A140319(n)=if(n,prime(n)^A140319(n-1),1) \\ M. F. Hasler, Nov 03 2009

Sum_{n>=0} 1/a(n) = A167155. - Amiram Eldar, Nov 15 2020

Corrected offset/definition, and initial a(0)=1 from M. F. Hasler, Nov 03 2009

A080219 Decimal expansion of exponential factorial constant Sum_{n>=1} 1/A049384(n).

Original entry on oeis.org

1, 6, 1, 1, 1, 1, 4, 9, 2, 5, 8, 0, 8, 3, 7, 6, 7, 3, 6, 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

Eric W. Weisstein, Feb 06 2003

This is a Liouville number and therefore transcendental.

			1/1 + 1/2 + 1/9 + 1/262144 + ... = 1.611114925808376736111...

Contributed by Jonathan Sondow.

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
Index entries for transcendental numbers

Cf. A049384, A167155.

eFac[1] = 1; eFac[n_] := eFac[n] = n^eFac[n-1]; Clear[s]; s[m_] := s[m] = RealDigits[Sum[1/eFac[n], {n, 1, m}], 10, 100] // First; s[m = 1]; While[s[m] != s[m - 1], m++]; s[m] (* Jean-François Alcover, Feb 08 2013 *)

cp's OEIS Frontend

A140319 Exponential primorial: a(n)=prime(n)^a(n-1), a(0)=1, where prime(n)=A000040(n).

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