Special Sums of Squares

There are many positive integers $n$ with the property that:

$$n=x^2+y^2$$ where $x$ and $y$ are integers but how frequent are integers with the additional property that $x$ and $y$ are both different but share the same digits. The first example of such a number is: $$585 = 12^2+ 21^2$$ In fact, up to 40000, there are 51 such numbers. They are:

585, 1130, 1553, 1877, 2340, 2826, 3005, 3329, 3977, 4034, 4520, 4941, 5265, 5330, 5913, 6212, 6698, 6885, 7361, 7508, 7685, 8333, 8642, 8874, 9305, 9360, 10170, 10265, 10589, 11237, 12020, 12506, 13653, 13977, 15650, 17525, 22301, 24804, 27185, 27509, 29930, 30416, 32553, 32877, 33525, 35540, 36026, 36836, 38405, 38729, 39377

Here is a permalink that will generate these numbers and their factorisations. The sequence is not listed in the OEIS. I was drawn to investigate the frequency of these sorts of numbers because the number associated with my diurnal age today, 27509, has this property:

$$27509=103^2+130^2$$ The number also has the property that the difference of 130 and 103 is 27, a cube, and this qualifies the number for membership of OEIS  A282405:

A282405

Primes $p = x^2 + y^2$ such that $x - y$ is a cube greater than one.

The initial members of the sequence are (permalink):

977, 1049, 1289, 1877, 2477, 2609, 3329, 4877, 5669, 6089, 6977, 8429, 9209, 9749, 10589, 12377, 12689, 13649, 15329, 15877, 16657, 17477, 18617, 18913, 19213, 20773, 21377, 21757, 22093, 22433, 22777, 23833, 23909, 25229, 25673, 26053, 26437, 27509, 30497

The first member of this sequence, 977, has the property that: $$977=31^2+4^2\\ \text{where } 31-4=27=3^3$$ Of course, the difference need not be a cubic number. It could be a square number. In such case, the numbers belong to OEIS A282406:

A282406

Primes $p = x^2 + y^2$ such that $x - y$ is a square greater than one.

The first member of the sequence is 101 with the property that: $$101=10^2-1^2\\ \text{where }10 -1 = 9 =3^2$$ This sequence of numbers in not in the OEIS. The 152 initial numbers, up to 40000, are (permalink):

101, 353, 461, 521, 653, 677, 733, 857, 881, 997, 1153, 1237, 1553, 1613, 1901, 2053, 2153, 2297, 2557, 2693, 2713, 2833, 3061, 3313, 3433, 3581, 3593, 4001, 4013, 4273, 4481, 4637, 4813, 5413, 5981, 6037, 6101, 6301, 6473, 6653, 7121, 7393, 7793, 7853, 7877, 8377, 8521, 8893, 9013, 9157, 9221, 9521, 9697, 9781, 9973, 10253, 10313, 10601, 10861, 11093, 11117, 12301, 12601, 12637, 12941, 12953, 13001, 13597, 13841, 14321, 14593, 14813, 15277, 15641, 15901, 16061, 16333, 16421, 16433, 16693, 16981, 17581, 18313, 18553, 18593, 19301, 19333, 19441, 19661, 19717, 19841, 19961, 20113, 20393, 21001, 21401, 21521, 21601, 21737, 21881, 22153, 22573, 23041, 23081, 23857, 24733, 25121, 25541, 25561, 25621, 26261, 26393, 26513, 26993, 27457, 27653, 27701, 28813, 28901, 29501, 29581, 29761, 29837, 30241, 30661, 30817, 30893, 31393, 31541, 31741, 32141, 32321, 32633, 33581, 33713, 34781, 34897, 35153, 36313, 36493, 36541, 36761, 36821, 37853, 38261, 38321, 38393, 38677, 38821, 39233, 39461, 39521

The algorithm is easily modified to accommodate other roots. We need not restrict ourselves to differences. What about sums? Let's consider:

Primes $p = x^2 + y^2$ such that $x + y$ is a square greater than one.

The first example of such a number is: $$53=2^2+7^2\\ \text{where }2+7=9=3^2$$ There are 83 such numbers in the range up to 40000. They are (permalink):

53, 317, 337, 353, 373, 397, 457, 577, 1213, 1381, 1621, 2213, 3461, 3593, 3701, 3761, 4481, 4793, 5021, 5393, 5801, 7333, 7433, 7541, 7741, 7933, 8081, 8161, 8521, 9181, 9433, 10133, 10601, 11833, 12421, 13933, 14293, 14321, 14341, 14401, 14461, 14593, 15121, 15581, 16141, 16661, 17093, 17401, 18793, 19181, 19381, 19793, 20441, 21601, 22093, 22861, 24793, 25373, 25457, 25577, 25733, 25793, 25997, 26153, 26237, 26293, 26417, 26513, 26717, 26921, 27241, 27893, 28277, 28433, 29453, 31253, 32633, 33377, 33893, 34157, 35537, 36713, 38273

Similarly we could consider numbers such as:

Primes $p = x^2 + y^2$ such that $x + y$ is a cube greater than one.

The first example of such a prime is: $$389=10^2+17^2\\ \text{where }10+17=27=3^3$$ The sequence of such numbers is not in the OEIS. There are 27 such numbers in the range up to 40000. They are (permalink):

389, 449, 509, 677, 7817, 7853, 7873, 7993, 8233, 8293, 8573, 8737, 9013, 9437, 10193, 10333, 10477, 11093, 11257, 11597, 11953, 12517, 12713, 13537, 14197, 14657, 15377