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%I #34 Sep 08 2022 08:45:14
%S 16,17,63,73,364,437,545,573,963,6475,23797,67458,2475989,2475998
%N Numbers n such that pi(n) = product of digits of n.
%C Next term is greater than 2*10^8. The only numbers with the property that pi(n) = sum of the digits of n, are the three numbers 15, 27 & 39.
%C a(15) > 10^13, if it exists. - _Giovanni Resta_, Sep 02 2017
%C a(15) > 10^15, if it exists. - _Chai Wah Wu_, Apr 23 2018
%C 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
%C 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
%e 2475998 is in the sequence because pi(2475998)=2*4*7*5*9*9*8.
%e 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
%t 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}]
%t Select[Range[2500000],PrimePi[#]==Times@@IntegerDigits[#]&] (* _Harvey P. Dale_, Dec 04 2012 *)
%o (PARI) isok(n) = primepi(n) == factorback(digits(n)); \\ _Michel Marcus_, Apr 23 2018
%o (Magma) [n: n in [1..10^5] | &*Intseq((n)) eq #PrimesUpTo(n)]; // _Vincenzo Librandi_, Nov 06 2018
%Y Cf. A002473, A007954, A000720.
%Y Cf. A097221, A097222, A097223.
%K base,more,nonn,fini
%O 1,1
%A _Farideh Firoozbakht_, Aug 02 2004
%E Keyword fini from _Jeppe Stig Nielsen_, Nov 04 2018
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