Problems & Puzzles: Puzzles

Puzzle 735. Consecutive primes such that the sum of their digits generates another set of consecutive primes. Here we ask for the earliest set of n consecutive primes P={p1, p2, ..., pn} that generates an associated set of primes Q={q1, q2, ..., qn} such that qi=SOD(pi) for all i=1 to n, and such that: a) The set Q is composed only by n consecutive primes in strict order (see OEIS, A239790) * b) The set Q is composed by n prime numbers, no specific order is needed (see OEIS, A240598) Q. Find some few next terms for each sequence? ________________ Based on the entry 1290 from the Claudio Meller´s site.

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Contributions came from J. K. Andersen

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Andersen wrote:

a) Here is a 24-digit upper limit for n=7:
101100010001001200110001 + d, for d = 0, 110, 222, 350, 372, 378, 398.
Q = {11, 13, 17, 19, 23, 29, 31}
It's probably not the earliest set. That looks hard to find.

Claudio Meller愀 page is based on
https://groups.yahoo.com/neo/groups/primenumbers/conversations/topics/25508
There I found a non-minimal set of 10 consecutive primes where the sums of
digits are also a set of 10 consecutive primes, but not in increasing order:
536220186773648241169499941 + d,
for d = 0, 22, 28, 48, 60, 82, 102, 106, 108, 148.
Q = {127, 131, 137, 139, 97, 101, 103, 107, 109, 113}.
The only non-increasing step is 139 to 97.

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