A164323 - OEIS

%I #21 Jun 23 2023 01:58:56

%S 15,383,629,8297

%N Numbers m such that m = prime(P) + phi(P), where P is the product of the digits of m.

%C The product of the digits of next term (if it exists) is greater than 2*10^8.

%C The sequence is finite since prime(P) ~= P*log(P) and phi(P) < P, while n > 10^(log_9(P)) - 1 > P^1.047. - _Max Alekseyev_, Dec 14 2011

%C By computation, any further terms must have P > 10^17.  By applying the inequalities p_k < k * (log(k) + log(log(k))) and P < 9^(1 + log_10(n/9)) to the defining equation, any further terms must have m < 1.29 * 10^45. - _Lucas A. Brown_, Jun 20 2023

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A164323.py">Python program</a>.

%e 8297 = prime(8*2*9*7) + phi(8*2*9*7), so 8297 is in the sequence.

%t Do[If[n=Prime[m]+Eulerphi[m];m==Apply[Times,IntegerDigits[n]],Print[n]],

%t {m,200000000}]

%t pdnQ[n_]:=Module[{p=Times@@IntegerDigits[n]},If[p>0,n==Prime[p]+ EulerPhi[ p],0]]; Select[Range[8300],pdnQ] (* _Harvey P. Dale_, Aug 12 2022 *)

%Y Cf. A000040, A000010, A007954, A164322, A164324.

%K base,more,nonn,fini

%O 1,1

%A _Farideh Firoozbakht_, Aug 13 2009

%E Keyword fini added by _Max Alekseyev_, Dec 14 2011