Problems & Puzzles: Puzzles

 Puzzle 1065 A larger integer than 45 such that... JM Bergot sent the following nice puzzle: The sum of the divisors of 45 is 78, with divisors 1, 3, 5, 9, 15, 45. 78mod1=0 78mod3=0 78mod5=3 78mod9=6 78mid15=3 78mod45=33 Add the remainders 0+0+3+6+3+33=45. Q. Can you find a larger number than 45 with this property?

During the week 27- 3 Dec., 2021, contributions came from Oscar Volpatti, Paolo Lava, Emmanuel Vantieghem, Giorgos Kalogeropoulos

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Oscar wrote:

I checked all numbers below 10^9, finding five larger solutions.
Here's the complete list:
4, 45, 6048, 14421, 26409026, 29270772, 30685402...

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Paolo wrote:

You can easily find the first 4 numbers: 4, 45, 6048 and 14421.

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Emmanuel wrote:

If I made no mistakes, the set of all the "solutions" less than 11*10^9 is :
{4, 45, 6048, 14421, 26409026, 29270772, 30685402}.

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Giorgos wrote:

These numbers are the fixed points of OEIS
The only numbers that I found having this property are:
4, 45, 6048, 14421, 26409026, 29270772, 30685402

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