Template:Sequence of the Day for August 23

Intended for: August 23, 2011

Timetable

• First draft entered by Alonso del Arte on May 28, 2011 as a verbatim copy of a write-up from December 23, 2010 ✓
• Draft reviewed by Alonso del Arte on August 21, 2011 ✓
• Draft approved by Daniel Forgues on August 23, 2011 ✓

Yesterday's SOTD * Tomorrow's SOTD

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A092447: Concatenate odd primes in decreasing order.*

{ 3, 53, 753, 11753, 1311753, 171311753, 19171311753, ... }

Quite self-explanatory. Observe that 3, 53 and 171311753 are odd prime “concatenation of odd primes in decreasing order.” Are there any more?

Yes, there are more: A092448 Primes in A092447.

a (1) = 3 (1 digit);

a (2) = 53 (2 digits);

a (3) = 171311753 (9 digits)

a (4) = 8983797371676159534743413731292319171311753 (43 digits);

a (5) starts with odd prime 383 (198 digits) (see A100003);

a (6) starts with odd prime 8831 (4202 digits) (see A100003);

a (7) starts with (odd prime > 28800th prime = 335033) (more than ? digits) (see A100003).

I’m hinting that this sequence is infinite... but I have no clue how one could come up with a proof! — DF

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* If we concatenate odd primes in increasing order (A089933), do we ever get primes (apart from 3, obviously)? Except for 3, these numbers are not prime for n < 10000.