Problems & Puzzles: Puzzles

Puzzle 500. 211896 Farideh Firoozbakht sent the following puzzle: 211896 = phi(211896/2) + phi(211896/1) + phi(211896/1) + phi(211896/8) + phi(211896/9) + phi(211896/6). Farideh also found other numbers like this one: 61341696, 141732864, 219483432, 1423392768, 4844814336 Q. What is the next such number?

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Contributions came from Mr. Qu Shun Liang & W Edwin Clark.

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Qu Shun Liang wrote:

I found 7 more terms!

16484622336
23362267824
28193299344
169699442688
993338339328
2344883866416
423732883488768

....

for 1<=k<=9, phi(n/k)=phi(n)/t, there 1<=t<=9. so, if n is a solution,
n=phi(n)*u/v, there u/v=s1*/1+s2*/2+s3*/3+…+s9/9.
so we get: actually, The largest prime factor of u is very small.
Further more, The largest prime factor of n is very small.
take n=61341696 for example, 61341696=2^19*3^2*13^1

I have search all number that the largest prime factor is less than 256 in [1,10^15]

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W. E. Clark wrote:

I think, but am not absolutely sure, that the next number in Farideh's
list is 16484622336.

I found 14 numbers satisfying Farideh's criterion: All of
them have the form 2^a*3^b*7^d*p^e where p is prime > 11. (This includes
the 6 already found by Farideh.) They are:

211896, 61341696, 141732864, 219483432, 1423392768, 4844814336,
16484622336, 23362267824, 28193299344, 169699442688, 993338339328,
2344883866416, 8829641374848, 423732883488768

But I make no claim that there aren't others in between.

Two questions: Are there any containing TWO distinct prime factors different from 2,3,7? Is there an example containing 5 as a factor?

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Fred answer to Clark's question:

No, here's why:

1)The totient(x) for every x except for 2 is an even number. The upper bound where for n where n/d = 2 (d is some digit) is 18. There are no solutions through 18. Therefore, there can be no odd solutions to this puzzle in general because the sum of even numbers can never be odd.
2) A number can only be divisible by 5 if its unit digit is 0 or 5.
3) An odd solution (ending in 5) is not possible as stated above.
4) An even solution ending in 0 is not possible for some n because phi(n/0) is undefined.

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Clark added:

Hi Farideh,

Concerning puzzle number 500: In case you are planning on submitting the sequence to the OEIS: I have now checked that there are no numbers satisfying your condition between 4844814336 and 16484622336. So you can safely add 16484622336 to the list. But I cannot speak about the lack of intermediate values for the other numbers in my list.

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