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A164323
Numbers m such that m = prime(P) + phi(P), where P is the product of the digits of m.
3
15, 383, 629, 8297
OFFSET
1,1
COMMENTS
The product of the digits of next term (if it exists) is greater than 2*10^8.
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
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
EXAMPLE
8297 = prime(8*2*9*7) + phi(8*2*9*7), so 8297 is in the sequence.
MATHEMATICA
Do[If[n=Prime[m]+Eulerphi[m]; m==Apply[Times, IntegerDigits[n]], Print[n]],
{m, 200000000}]
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 *)
CROSSREFS
KEYWORD
base,more,nonn,fini
AUTHOR
Farideh Firoozbakht, Aug 13 2009
EXTENSIONS
Keyword fini added by Max Alekseyev, Dec 14 2011
STATUS
approved