Super Sphenic Numbers

The number associated with my diurnal age today (28262) is what might be termed a "super sphenic number" as I'll explain in this post. Firstly however, its factorisation:$$28262=2 \times 13 \times 1087$$If we reverse its digits, we get the number 26282 and this number is also sphenic:$$26282 = 2 \times 17 \times 773$$Let's now concatenate the factors of 28262 in ascending order. This gives us the number 2131087 which is also sphenic:$$2131087 = 7 \times 167 \times 1823$$28262 has a sum of digits of 20 and if we add this to the original number we get the palindromic number 28282 which is also sphenic:$$28262+20=28282 = 2 \times 79 \times 179$$The number has a product of digits of 384 and if we subtract this from the original number we get 27878 which is sphenic:$$28262 - 384 = 27878 = 2 \times 53 \times 263$$If we consider only the internal digits of 28262, we get the number 826 which is also sphenic:$$2\, 826 \, 2 \rightarrow826=2 \times 7 \times 59$$When a sphenic number is considered as a sphenic brick then it has an associated number in the form of the brick's surface area. In the case of 28262, this associated surface area is 32662 square units and this number too is sphenic:$$32662 = 2 \times 7 \times 2333$$28262 has a sum of proper divisors that is also sphenic:$$ \sigma(28262) - 28262 =17434 = 2 \times 23 \times 379 $$The number has a sum of prime factors (1102) that is sphenic:$$2 +13+1087=1102 = 2 \times 19 \times 29$$28262 has a totient of 13032 which is not sphenic but its cototient (number - totient) of 15230 is:$$28262 - 13032 = 15230 = 2 \times 5 \times 1523$$So we can see that 28262 may well be termed a super sphenic number because of the above associations.

Concatenating Reversible Sphenic Numbers

My previous post Concatenating Emirpimes suggested the concatenation of reversible sphenic numbers as a logical sequel. A sphenic number that is still sphenic when its digits are reversed is sometimes called a \( \textbf{cinephs} \). Let's take 418 as an example of the type of number that we want to identify:$$ \begin{align} 418 &= 2 \times 11 \times 19 \\ 814 &= 2 \times 11 \times 37 \\ 418814 &= 2 \times 11 \times 19037 \end{align}$$There are, coincidentally, 418 such numbers in the range up to 40000. Here are the details (permalink):

=== Cineph Concatenation Statistics ===

Range evaluated: 1 to 40000

1. Sphenic numbers: 7720 (19.30% of the range)

2. Cinephs: 1890 (4.72% of the range)

3. Successful Cinephs: 418 (1.0450% of the range)

=== Comma-Separated List of Successful Cinephs ===

418, 682, 759, 814, 957, 1034, 1095, 1113, 1185, 1342, 1419, 1446, 1606, 1614, 1902, 1965, 2014, 2035, 2282, 2414, 2431, 2438, 2494, 2635, 2665, 2686, 3018, 3059, 3138, 3201, 3278, 3297, 3358, 3495, 3597, 3606, 3678, 3685, 3714, 3729, 3765, 3926, 4301, 4382, 4543, 4565, 4715, 4945, 5174, 5362, 5495, 5863, 5986, 6035, 6061, 6083, 6206, 6293, 6806, 7185, 7305, 7414, 7449, 7567, 7645, 7657, 7662, 7718, 8155, 8174, 8378, 8386, 8390, 8393, 8555, 8734, 8786, 8789, 8987, 9138, 9141, 9213, 9309, 9321, 9354, 9362, 9503, 9515, 9519, 9807, 9822, 9885, 9951, 9978, 10095, 10246, 10263, 10274, 10326, 10382, 10490, 10502, 10509, 10554, 10635, 10761, 10835, 10879, 10915, 11098, 11342, 11395, 11398, 11407, 11577, 11605, 11753, 11922, 11937, 11958, 11982, 12174, 12218, 12257, 12378, 12498, 12529, 12551, 12595, 12859, 12874, 12890, 12914, 13137, 13143, 13222, 13305, 13514, 13574, 13618, 13634, 13783, 13786, 13906, 14035, 14043, 14055, 14127, 14174, 14223, 14298, 14313, 14443, 14573, 14590, 14619, 14646, 14655, 14846, 14889, 14894, 14955, 15235, 15323, 15422, 15430, 15521, 15585, 15626, 15657, 15794, 15818, 15914, 15958, 15994, 16098, 16159, 16518, 16558, 16666, 16701, 16914, 16915, 16946, 17006, 17139, 17353, 17445, 17518, 17534, 17799, 17805, 17814, 17818, 17978, 18093, 18123, 18147, 18205, 18222, 18231, 18309, 18395, 18458, 18555, 18579, 18685, 18717, 18814, 18854, 19023, 19178, 19221, 19317, 19382, 19545, 19605, 19623, 19877, 19923, 19947, 19987, 20009, 20066, 20195, 20270, 20305, 20315, 20530, 20594, 20657, 20726, 20758, 20870, 20905, 20945, 20965, 20966, 22058, 22282, 22562, 22582, 22591, 22645, 22933, 22939, 22945, 22970, 24002, 24026, 24095, 24265, 24409, 24479, 24583, 24590, 24739, 24817, 24878, 24962, 24973, 26090, 26102, 26134, 26146, 26245, 26266, 26273, 26429, 26494, 26506, 26546, 26555, 26614, 26638, 26663, 26758, 26873, 26878, 26930, 26939, 26990, 28085, 28165, 28186, 28222, 28237, 28358, 28418, 28426, 28441, 28483, 28514, 28645, 28747, 28835, 28870, 28918, 30054, 30099, 30126, 30277, 30418, 30502, 30651, 30795, 30943, 31026, 31027, 31035, 31191, 31254, 31386, 31658, 31719, 31978, 32039, 32043, 32181, 32295, 32351, 32417, 32442, 32613, 32846, 32857, 32898, 32907, 32938, 33143, 33162, 33186, 33378, 33429, 33473, 33555, 33583, 33585, 33754, 33818, 33835, 33906, 33918, 33922, 34005, 34077, 34078, 34131, 34133, 34158, 34203, 34222, 34257, 34474, 34562, 34563, 34661, 34705, 34826, 35006, 35094, 35123, 35202, 35274, 35319, 35382, 35618, 35619, 35871, 35949, 35985, 36003, 36039, 36165, 36174, 36226, 36355, 36474, 36586, 36606, 36627, 36935, 36958, 37317, 37329, 37367, 37411, 37418, 37433, 37527, 37614, 37634, 37743, 37754, 37970, 38049, 38085, 38193, 38294, 38399, 38451, 38467, 38533, 38555, 38643, 39081, 39169, 39305, 39306, 39351, 39358, 39399, 39458, 39562, 39785, 39842, 39918, 39966

As we learned in my previous post: 

There is a fundamental rule in number theory: Every palindrome with an even number of digits is divisible by 11.

Thus all the above numbers have 11 as a prime factor. Let's take 39358 as another example: $$ \begin{align} 39358 &= 2 \times 11 \times 1789 \\ 85393 &= 7 \times 11 \times 1109 \\ 3935885393 &= 11 \times 397 \times 901279\end{align}$$Of course, if we were to try this technique on emirps (reversible primes), we would never end up with a prime because the resultant number would always be divisible by 11. For example, 37 and 73 combine to form$$ 3773 = 11 \times 343 =11 \times 7^3$$

Reversible Sphenic Numbers

There is the reversible prime, known as an emirp. There is the reversible semiprime, known as an emirpimes, and then there is the reversible sphenic number, known as an cinehps. This is a rather ugly term so I'll just use the term reversible sphenic number. An example of an emirp is 17 whose reversal, 71, is also prime. An example of an emirpimes is 26 = 2 x 13 whose reversal, 62 = 2 x 31, is also a semiprime. The first example of a cinehps is 165 = 3 x 5 x 11 whose reversal, 561 = 3 x 11 x 17, is also a sphenic number.

These numbers form OEIS A270175:

A270175

Cinehps numbers: sphenic numbers whose reversal is a different sphenic number.

Note that palindromic sphenic numbers are excluded. The initial members of the sequence are:

165, 246, 285, 286, 366, 418, 435, 438, 498, 534, 561, 582, 609, 642, 663, 682, 759, 814, 834, 894, 906, 957, 1002, 1023, 1034, 1066, 1095, 1113, 1131, 1185, 1209, 1239, 1245, 1265, 1311, 1342, 1353, 1374, 1398, 1419, 1443, 1446, 1479, 1515, 1526, 1542, 1545, ...

Up to the one million mark, these numbers total 5.28% of the range. One of the concepts associated with a sphenic number is that of the sphenic brick. Let's consider a sphenic number \(n\) whose factors are \(a,b,c\). The sphenic brick is the three dimensional cuboid with volume of \(n\) cubic units and linear dimensions of \(a,b\) and \(c\) units.

Such a brick has an associated surface area and a thought occurred to me. Are there any reversible sphenic number pairs that each have the same surface area? A little investigation revealed that there are. They are rare birds indeed however, and there are only eight ( in four pairs) in the range up to three million (366 and 663, 3245 and 5423, 3685 and 5863, 921239 and 932129). Here are the details and here is the permalink to the calculation. Surface areas are shown in bold red.

366  = 2 x 3 x 61 --> 622 and 663 = 3 x 13 x 17 --> 622

3245 = 5 x 11 x 59 --> 1998 and 5423 = 11 x 17 x 29 --> 1998

3685 = 5 x 11 x 67 --> 2254 and 5863 = 11 x 13 x 41 --> 2254

921239 = 11 x 89 x 941 --> 190158 and 932129 = 11 x 101 x 839 --> 190158

None of the sphenic bricks associated with these numbers look like bricks because they are all very elongated but that's the term that is used for these shapes. Figure 1 shows that the 11 x 89 x 941 looks more like a plank than a brick.

Figure 1

There may well be more beyond the three million mark but SageMathCell timed out above that. Anyway, fascinating that such numbers exist with the first of them being 366, the number of days in a leap year. It can be noted that 366 is the only even number. Placed in sequence on the number line we have:

366, 663, 3245, 3685, 5423, 5863, 921239, 932129

This sequence of terms could be described thus:

Non-palindromic sphenic numbers which, when reversed, are also sphenic numbers with the members of both pairs having identical sphenic brick surface areas. 

Not surprisingly this sequence does not appear in the OEIS, nor will it, as I have ceased to contribute.

ADDENDUM:

The above idea can be applied to semiprimes as well. In this case we will be working with two dimensional rectangles. The two factors of the semiprime form the length and width of the associated rectangle. In the range up to one million, reversible semiprimes comprise 6.06% of the total range but there are only 26 reversible semiprimes with the property that their associated areas are equal. See permalink for calculation.

The pairs (up to one million) are:

14269 = 19 x 751 --> 1540 and 96241 = 157 x 613 --> 1540

15167 = 29 x 523 --> 1104 and 76151 = 271 x 281 --> 1104

16237 = 13 x 1249 --> 2524 and 73261 = 61 x 1201 --> 2524

18449 = 19 x 971 --> 1980 and 94481 = 107 x 883 --> 1980

18977 = 7 x 2711 --> 5436 and 77981 = 29 x 2689 --> 5436

36679 = 43 x 853 --> 1792 and 97663 = 127 x 769 --> 1792

140941 = 97 x 1453 --> 3100 and 149041 = 103 x 1447 --> 3100

150251 = 347 x* 433 --> 1560 and 152051 = 383 x 397 --> 1560

196891 = 401 x 491 --> 1784 and 198691 = 431 x 461 --> 1784

302363 = 211 x 1433 --> 3288 and 363203 = 263 x 1381 --> 3288

308459 = 173 x 1783 --> 3912 and 954803 = 937 x 1019 --> 3912

319853 = 317 x 1009 --> 2652 and 358913 = 379 x 947 --> 2652

958099 = 761 x 1259 --> 4040 and 990859 = 839 x 1181 --> 4040

It can be noted that all these reversible semiprimes are odd.

If we are even more adventurous we can consider numbers with four distinct factors as being associated with 4-dimensional hypercuboids. When unfolded into 3-dimensions, the eight cuboids are the equivalent of the surface area of our sphenic bricks. In the range up to three million, about 1.48% of numbers are reversible four factor numbers with the first one being 1518 = 2 x 3 x 11 x 23 and 8151 = 3 x 11 x 13 x 19. However, no number pairs emerge with identical volumes of unfolded cuboids. Figure 1 shows an unfolded hypercube and an unfolded hypercuboid will look similar but will have four pairs of cuboids each with different volumes, instead of consisting of eight identical cubes. Permalink (which may need double checking).

Figure 1