Prime that is sum of consecutive triangular numbers with prime indices — Part 1

Paul asks:

If we consider the Index numbers 29 and 43, both prime and sum the consecutive triangular numbers with those prime indices the sum is the prime number 176809, there are many such cases with a start and end prime indices, in this case the number of primes in the range inclusive {29, 31, 37, 41, 43} is also a prime number, there are not so many of those and in the case of {3, 7, 353} where 3 and 7 are the indices and 373 is the sum then we find 373 is a palindromic prime, find the next such case.

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