A066457 Numbers k such that product of factorials of digits of k equals pi(k) (A000720).

13, 1512, 1520, 1521, 12016, 12035, 226130351, 209210612202, 209210612212, 209210612220, 209210612221, 13030323000581525

Offset: 1

Author

Jason Earls, Jan 02 2002

Keywords

• nonn
• base
• more

Comments

The Caldwell/Honaker paper does not discuss this, only suggests further areas of investigation.

There are no other members of the sequence up to and including n=1000000. - Harvey P. Dale, Jan 07 2002

If 10n is in the sequence and 10n+1 is composite then 10n+1 is also in the sequence (the proof is easy). - Farideh Firoozbakht, Oct 24 2008

a(13) > 10^19 if it exists. - Chai Wah Wu, May 03 2018

Examples

			12016 is a term because there are exactly 1!*2!*0!*1!*6! (or 1440) prime numbers less than or equal to 12016.
pi(209210612202) = 8360755200 = 2!*0!*9!*2!*1!*0!*6!*1!*2!*2!*0!*2!. [Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008]
		

Links

• C. Caldwell and G. L. Honaker, Jr., Is pi(6521)=6!+5!+2!+1! unique?
• A discussion about this topic: bbs.emath.ac.cn [From Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008]
• Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Crossrefs

Cf. A000720, A066459, A049529, A105327.

Programs

Mathematica

Select[Range[1000000], Times@@( # !&/@IntegerDigits[ # ])==PrimePi[ # ]&]

PARI

isok(n) = my(d = digits(n)); prod(k=1, #d, d[k]!) == primepi(n); \\ Michel Marcus, May 04 2018

Extensions

a(7) from Farideh Firoozbakht, Apr 20 2005

a(8)-a(11) from Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008

a(12) from Chai Wah Wu, May 03 2018