What's Special About 516493?

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I got to thinking about primes that remain prime under the operation of prime + digit sum. The acronym "sod" standing for sum of digits is commonly used in this context. There are many primes that satisfy. For example, 11 has a digit sum of 2 and 11 + 2 = 13 which is prime. If we repeat the operation we get 13 + 4 = 17 which is still prime. However, repeating the operation again yields 17 + 8 = 25 which is composite.

The primes that remain prime for only one application of the prime + sod operation constitute OEIS A048523 (permalink):

A048523

Primes for which only one iteration of 'Prime plus its digit sum equals a prime' is possible.

The initial members are:

13, 19, 37, 53, 71, 73, 97, 103, 127, 163, 181, 233, 271, 307, 383, 389, 431, 433, 499, 509, 563, 587, 631, 701, 743, 787, 811, 857, 859, 947, 1009, 1049, 1061, 1087, 1153, 1171, 1223, 1283, 1423, 1483, 1489, 1553, 1597, 1601, 1607, 1733, 1801, 1861, 1867

There are 7939 such primes in the range up to one million. Notice that 11 is not included here because it produces a prime again. The primes, like 11, that remain prime for only two applications of the prime + sod operation constitute OEIS A048524 (permalink):

A048524

Primes for which only two iterations of 'Prime plus its digit sum equals a prime' are possible.

The initial members of the sequence are:

11, 59, 101, 149, 167, 257, 293, 367, 419, 479, 547, 617, 727, 839, 1409, 1579, 1847, 2039, 2129, 2617, 2657, 2837, 3449, 3517, 3539, 3607, 3719, 4217, 4637, 4877, 5689, 5807, 5861, 6037, 6257, 6761, 7027, 7517, 8039, 8741, 8969, 9371, 9377, 10667

There are 1111 such primes in the range up to one million. The primes that remain prime for only three applications of the prime + sod operation constitute OEIS A048525 (permalink):

A048525

Primes for which only three iterations of 'Prime plus its digit sum equals a prime' are possible.

The initial members of the sequence are:

277, 1559, 5779, 7489, 11279, 15091, 22093, 37811, 43579, 46279, 48541, 49957, 53479, 54751, 60589, 68473, 72883, 74821, 83621, 85621, 90793, 91921, 93901, 97501, 107981, 110899, 111799, 120193, 153379, 157739, 170299, 180731, 184441

There are 136 such primes in the range up to one million. The primes that remain prime for only four applications of the prime + sod operation constitute OEIS A048526 (permalink):

A048526

Primes for which only four iterations of 'Prime plus its digit sum equals a prime' are possible.

The initial members of the sequence are:

37783, 85601, 259631, 268721, 350941, 371939, 378901, 516521, 665111, 733331, 883331, 967781, 1047929, 1056521, 1081721, 1258811, 1427411, 1480573, 1515929, 1584901, 1614929, 1842131, 1875311, 1885981, 2027801, 2044873, 2450531

There are only 12 such primes in the range up to one million. The primes that remain prime for only five applications of the prime + sod operation constitute OEIS A048527 (permalink):

A048527

Primes for which only five iterations of 'Prime plus its digit sum equals a prime' are possible.

The initial members of the sequence are:

516493, 1056493, 1427383, 1885943, 3166183, 3805183, 4241593, 6621283, 7646953, 12912283, 17987839, 32106493, 107152093, 120224773, 131144473, 133210873, 139388891, 142782877, 150326173, 155382923, 177865819, 184081943, 227795839, 242376877, 264174877

There is only one such prime in the range up to one million and that is 516493 and that is why it is deserving of this special post. The progression is shown below where True = Prime and False = Composite:

516493 True

516521 True

516541 True

516563 True

516589 True

516623 True

516646 False

516493 is the only prime in the range up to one million that generates five successive primes under the prime + sum of digits operation

There are primes that go further than five iterations. For example, 286330897 survives seven iterations (link). This is the progression:

286330897 True

286330943 True

286330981 True

286331021 True

286331047 True

286331081 True

286331113 True

286331141 True

286331170 False

The number 56676324799 survives eight iterations (link). This is the progression:

56676324799 True

56676324863 True

56676324919 True

56676324977 True

56676325039 True

56676325091 True

56676325141 True

56676325187 True

56676325243 True

56676325292 False

Presumably there is no limit to the number of iterations.