Semiprimes Within Semiprimes

I got to thinking about semiprimes and how many of them have the property that the digits of their two prime factors can both be rearranged to form two new semiprimes. It's as if two additional semiprimes are hiding within the factors of the original semiprime.

It's simple enough to investigate and I could have written the code but I lazily got Gemini to do it for me. It turns out that there are 1077 such numbers in the range up to 40000 (permalink). the first such number is 361:$$ \begin{align} 361 &=19 \times 19 \\19 &\rightarrow 91 =7 \times 13 \end{align}$$The next such number is not square and is 779:$$ \begin{align} 779 &= 19 \times 41 \\ 19 &\rightarrow 91 =7 \times 13 \\ 41 &\rightarrow 14 =2 \times 7 \end{align}$$Here is a list of the numbers from 28247 (my diurnal age tomorrow) up to 40000:

28247, 28253, 28261, 28291, 28331, 28337, 28367, 28417, 28423, 28459, 28481, 28529, 28667, 28673, 28709, 28733, 28757, 28801, 28811, 28841, 28891, 28907, 28937, 28939, 28943, 28967, 28969, 28991, 28999, 29083, 29089, 29093, 29111, 29143, 29149, 29177, 29299, 29317, 29321, 29329, 29353, 29369, 29431, 29441, 29479, 29503, 29507, 29521, 29539, 29591, 29621, 29657, 29677, 29713, 29747, 29773, 29797, 29839, 29849, 29893, 29929, 29987, 30001, 30031, 30077, 30127, 30157, 30179, 30221, 30227, 30263, 30299, 30301, 30343, 30353, 30409, 30461, 30463, 30533, 30571, 30607, 30629, 30647, 30691, 30739, 30761, 30791, 30799, 30847, 30857, 30913, 30917, 30929, 30967, 30973, 31067, 31103, 31111, 31133, 31201, 31313, 31373, 31429, 31439, 31457, 31459, 31483, 31529, 31597, 31621, 31631, 31673, 31693, 31711, 31747, 31777, 31819, 31831, 31853, 31861, 31877, 31897, 31919, 31921, 31937, 31949, 32041, 32101, 32111, 32167, 32171, 32231, 32239, 32243, 32267, 32273, 32281, 32287, 32293, 32387, 32399, 32471, 32477, 32489, 32639, 32651, 32677, 32699, 32701, 32723, 32737, 32743, 32761, 32807, 32863, 32899, 32927, 32951, 33017, 33043, 33067, 33079, 33127, 33193, 33217, 33221, 33239, 33251, 33307, 33389, 33401, 33421, 33443, 33463, 33499, 33571, 33661, 33689, 33743, 33763, 33793, 33841, 33877, 33907, 33953, 33973, 33989, 33991, 34079, 34081, 34093, 34117, 34121, 34163, 34189, 34219, 34271, 34291, 34387, 34393, 34399, 34417, 34547, 34553, 34571, 34579, 34609, 34633, 34637, 34717, 34733, 34777, 34789, 34873, 34889, 34921, 34927, 34933, 34943, 34973, 34987, 35033, 35093, 35137, 35183, 35209, 35219, 35237, 35263, 35297, 35303, 35341, 35359, 35383, 35389, 35459, 35473, 35549, 35561, 35587, 35611, 35647, 35657, 35663, 35669, 35687, 35701, 35723, 35741, 35767, 35773, 35813, 35881, 35891, 35909, 35939, 35941, 35947, 35953, 35957, 36031, 36077, 36079, 36089, 36119, 36121, 36143, 36167, 36199, 36203, 36233, 36331, 36347, 36359, 36367, 36391, 36403, 36437, 36481, 36503, 36521, 36557, 36581, 36623, 36679, 36689, 36727, 36851, 36853, 36863, 36937, 36977, 36989, 37031, 37069, 37109, 37127, 37211, 37229, 37249, 37327, 37351, 37391, 37399, 37459, 37487, 37523, 37601, 37627, 37669, 37711, 37753, 37769, 37801, 37819, 37823, 37837, 37867, 37883, 37901, 37931, 37937, 37943, 37969, 37979, 37981, 38021, 38041, 38089, 38107, 38117, 38141, 38173, 38191, 38209, 38243, 38263, 38293, 38323, 38417, 38429, 38477, 38497, 38513, 38527, 38551, 38581, 38587, 38681, 38741, 38761, 38771, 38807, 38809, 38827, 38869, 38881, 38911, 38957, 38963, 38989, 38999, 39007, 39037, 39059, 39073, 39131, 39167, 39173, 39187, 39197, 39283, 39311, 39337, 39379, 39407, 39433, 39539, 39577, 39617, 39647, 39653, 39691, 39707, 39713, 39751, 39757, 39803, 39811, 39881, 39913, 39943, 39947

It was quite serendipitous that I thought about these sorts of numbers because, as I said, I'm 28247 days old tomorrow and this number has the property that:$$ \begin{align} 28247 &= 47 \times 601 \\47 &\rightarrow 74 =2 \times 37 \\ 601 &\rightarrow 106 =2 \times 53 \end{align}$$Of course this idea can be extended to sphenic numbers where each the three factors have digits that can be rearranged to form three new sphenic numbers. However, these numbers are comparatively speaking rather large. The first is 1113121 (permalink):$$ \begin{align} 1113121 &= 101 \times 103 \times 107 \\ 101&\rightarrow 110 =2 \times 5 \times 11 \\ 103 &\rightarrow 130 =2 \times 5 \times 13 \\ 107 &\rightarrow 170 = 2 \times 5 \times 17 \end{align}$$Gemini gives a good explanation of why the numbers need to so large (link).