A097220 Numbers n such that pi(n) = product of digits of n.

16, 17, 63, 73, 364, 437, 545, 573, 963, 6475, 23797, 67458, 2475989, 2475998

Offset: 1

Author

Farideh Firoozbakht, Aug 02 2004

Keywords

• base
• more
• nonn
• fini

Comments

The only numbers with the property that pi(n) = sum of the digits of n, are the three numbers 15, 27 & 39.

When n exceeds approximately 10^44, then pi(n) is consistently greater than the product of digits of n. So no term of this sequence exceeds 10^44. In particular, this sequence is finite. - Jeppe Stig Nielsen, Nov 04 2018

Products of digits of terms are in A002473. Term by term up to some bound (such that the bounds on primes hold), one could check terms t in A002473 on some known bounds. See example below. - David A. Corneth, Nov 06 2018

There are no other terms below 10^17. - Max Alekseyev, Nov 07 2024

Examples

			2475998 is in the sequence because pi(2475998)=2*4*7*5*9*9*8.
1152 is in A002473. As 8643 <= prime(1152) <= 9794. Examples of the 13 numbers with product of digits is 1152 in that interval are: 8944, 9288, 9448, 9484 none of which are terms. - _David A. Corneth_, Nov 06 2018
		

Crossrefs

Cf. A002473, A007954, A000720.

Cf. A097221, A097222, A097223.

Programs

Magma

[n: n in [1..10^5] | &*Intseq((n)) eq #PrimesUpTo(n)]; // Vincenzo Librandi, Nov 06 2018

Mathematica

v={}; Do[If[h=IntegerDigits[n]; l=Length[h]; p=Product[h[[k]], {k, l}]; PrimePi[n]==p, v=Append[v, n]; Print[v], If[Mod[n, 1000000]==0, Print[ -n]]], {n, 200000000}]
Select[Range[2500000],PrimePi[#]==Times@@IntegerDigits[#]&] (* Harvey P. Dale, Dec 04 2012 *)

PARI

isok(n) = primepi(n) == factorback(digits(n)); \\ Michel Marcus, Apr 23 2018

Extensions

Keyword fini from Jeppe Stig Nielsen, Nov 04 2018