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