Prime Indexed Primes

It struck me today that the indexing of primes is base-independent. It's obvious I know but for some reason I'd maintained this vague notion that the indexing was base-dependent. Today I turned 28103 days old and this number is the product of the 37th prime (157) and the 41st prime (179).

28103 is a member of OEIS A119658:

 A119658: area of consecutive prime-indexed prime rectangles or the product of consecutive prime-indexed primes.

The initial sequence members are:

15, 55, 187, 527, 1271, 2419, 3953, 5561, 9047, 13843, 19939, 28103, 34189, 40301, 50851, 66757, 78391, 93673, 116843, 129551, 147167, 172831, 198691, 234649, 278423, 307961, 330481, 351613, 369583, 437453, 523951, 571247, 616081, 684623, ...

As can be seen, the sequence members get large rather quickly. There are 575 prime-indexed primes in the range up to 40000 and they can be combined in various ways to generate new sequences such as OEIS A119658. Another example involving the sum instead of the product is OEIS A299644:

 

 A299644: a(\(n\)) = prime(prime(\(n+1\))) + prime(prime(\(n\))).

The initial sequence members are:

8, 16, 28, 48, 72, 100, 126, 150, 192, 236, 284, 336, 370, 402, 452, 518, 560, 614, 684, 720, 768, 832, 892, 970, 1056, 1110, 1150, 1186, 1216, 1326, 1448, 1512, 1570, 1656, 1736, 1796, 1886, 1958, 2022, 2094, 2150, 2240, 2324, 2372, 2418, 2514, 2706, 2842, 2880, 2918, ...

Another example is OEIS A318294 involving Sophie Germain primes:

 A318294: prime-indexed primes \(p\) such that \(2p + 1\) is also a prime-indexed prime.

The initial sequence members are:

5, 41, 1031, 3299, 3761, 6323, 7649, 7841, 10589, 12959, 18443, 37853, 38039, 44729, 58511, 59723, 60509, 77813, 105971, 134489, 137999, 190523, 194933, 200351, 239633, 250751, 266093, 288653, 315011, 366161, 376241, 377633, 393401, 410999, 420419, 420779, 450431, 474983

You can of course create sequences of your own that aren't in the OEIS database. For example, though the sum of two prime-indexed primes is always a composite number, what if we add 1 and ask whether the result is also a prime-indexed prime. The result is the following sequence which has 12 members below 40000:

17, 127, 1217, 3761, 4091, 5701, 8287, 10663, 21149, 35023, 37061, 39733

The first member is \(17_{\small{7} }= 5_{\small{3 }}+ 11_{\small{5 }}+ 1\) where the subscripts denote the prime index.