Problems & Puzzles: Puzzles

Puzzle 346. Happy 2006          First of all receive my best wishes, and even more, along this new year, 2006. Here you are invited to discover curio properties about the number 2006. Just to warm up the session, I introduce two of them, sent by Farideh Firoozbakht a few days ago: 1. 2006 is the only number n such that phi(phi(n))-pi(n)=phi(pi(n))=(d_1*d_k)^2 where d_1 d_2 ... d_k is the decimal expansion of n.   phi(phi(2006))-pi(2006)=phi(pi(2006))=(2*6)^2     2. 2006 is the only number n such that pi(pi(n))=a_k.a_1 and phi(pi(n))=(a_1*a_k)^2 where a_1 a_2 ... a_k is the decimal expansion of n and dot between numbers means concatenation.   pi(pi(2006))=62 & phi(pi(2006))=(2*6)^2 Q1. Send more 2006 numerical curios.

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Contributions came from Patrick Capelle & Farideh Firoozbakht.

Patrick wrote:

1. 2006²+1  and  2006²+3  are both prime. (2²+0²+0²+6²) +1  and  (2²+0²+0²+6²) +3  are both prime. It's the smallest four-digit number to have these two properties at the same time.

2. 2006 = 17² + (1717)

Expression using prime numbers (2 and 17), and a concatenation of 17 with 17. Note that if we delete the two zeros in 2006, we obtain : 26 = 22 + (2²)

3. sigma(2006) = sigma(2006 + (2+0+0+6)). 

4. The digit sum of the 2006th prime is prime. But it's also the case when we delete 1 or 2 zeros.

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Farideh wrote:

2006* (6002+1)^pi(2006) + prime(2006) is a titanic probable prime

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Pierandrea Formusa wrote on Feb 16, 2019

We can express 2006 like the sum of an even number n of distinct prime squares if and only if n is equal to six (e.g. 2006=3^2+7^2+11^2+13^2+17^2+37^2).

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