Problems & Puzzles: Puzzles

Puzzle 494. f(n)=product(prime(di)) Farideh Firoozbakht sent the following puzzle: The semiprime 185617 has this nice property:  phi(185617) = prime(1)*prime(8)*prime(5)*prime(6)*prime(1)*prime(7) Q1. Can you find another such number? Let's generalize the Farideh's model. Q2. Find other interesting solutions for other arithmetical functions f(n) such that: f(n)=product(prime(di)) where di are the digits of n in decimal notation.

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Contribution came from Jacques Tramu.

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Jacques wrote:

I have plotted Farideh function here : (with prime(0) = 0)

http://www.echolaliste.com/gbnums/p494.jpg

Q2. Experimented with different f(n) functions.
Results : n , f(n)

* f(n) = n
14 14
154 154
1196 1196
279174 279174

* f(n) = n/2
12 6
42 21

* f(n) = n-1
2692 2691
47482 47481

* f(n) = n+1
1 2
2 3
34 35
39444 39445
8653644 8653645

* f(n) = sum of prime factors (n)
5321 330
12228 1026
1614227 55692
2441142 12348

* f(n) = sigma(n) = sum of divisors(n)
2 3
4 7
148 266
484 931

* f(n) = pi(n) = numbers of primes <= n
123 30
5224 693
11166 1352
51174 5236
172451 15708
546322 45045
14355351 931700

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