Strong primes in cryptography are primes that are difficult to factor but in number theory a strong prime \(p_n\) has the property that:$$p_n>\frac{p_{n-1}+p_{n+1}}{2}$$If a sphenic number has three strong primes then it might be termed "triple strength" which is just a term that I came up with. These sorts of sphenic number constitute OEIS A363782:
A363782 | Products of three distinct strong primes. |
5423, 6919, 7667, 11033, 11803, 12529, 13079, 13277, 14773, 16687, 18139, 18241, 18821, 18887, 20009, 20213, 21373, 22649, 23749, 24013, 25201, 25619, 25789, 26609, 27269, 27863, 28897, 29087, 30217, 30481, 30943, 32021, 32153, 32219, 33031, 33473, 34133, 35003, 35629, 35717, 36839, 37111, 38947, 39479
Take 5423 = 11 x 17 x 29 as an example. 11 is closer to 13 than it is to 7, 17 is closer to 19 than it is to 13 and 29 is closer to 31 than it is to 23.
By contrast, sphenic numbers in which all three primes are weak are more numerous. A weak prime \(p_n\) has the property that:$$p_n<\frac{p_{n-1}+p_{n+1}}{2}$$These are not listed in the OEIS but there are 161 in the range up to 40,000 and they are (permalink):
1729, 2093, 2821, 3059, 3913, 4123, 4277, 4991, 5551, 5681, 5719, 6251, 6643, 6923, 7553, 7567, 7657, 8099, 8113, 9269, 9331, 9373, 9709, 9821, 9919, 10199, 10283, 10621, 11039, 11609, 11753, 11837, 11921, 12649, 12857, 13237, 13363, 13547, 13699, 13741, 14053, 14147, 14329, 14497, 15029, 15067, 15197, 15841, 16471, 16583, 17329, 17423, 17549, 17563, 18011, 18031, 18109, 18193, 18239, 18361, 18487, 18791, 18941, 19313, 20069, 20083, 20501, 20539, 20839, 21091, 21203, 21827, 21931, 21973, 21983, 22211, 22351, 22379, 23653, 24017, 24073, 24311, 24521, 24583, 24661, 24817, 24983, 25327, 25441, 25669, 25753, 26273, 26467, 26611, 26657, 26663, 26789, 26887, 26923, 27307, 27683, 27911, 28427, 28483, 28847, 29141, 29281, 29419, 30163, 30457, 30659, 30667, 30797, 30989, 31003, 31073, 31171, 31759, 31901, 32039, 32053, 32123, 32357, 32591, 32669, 32767, 32809, 33449, 33511, 33787, 33887, 34013, 34099, 34333, 34853, 35399, 35441, 35861, 35867, 35929, 36043, 36239, 36271, 36491, 36869, 37177, 37219, 37271, 37297, 37513, 37639, 38003, 38311, 38399, 38801, 38893, 38969, 39169, 39277, 39403, 39431
It should be noted that the famous taxi cab number, 1729, is the first member of this sequence and it factorises as follows:$$1729=7 \times 13 \times 19$$With 1729, we see that 7 is weak because it's closer to 5 than 11, 13 is weak because it's closer to 11 than 17 and 19 is weak because it's closer to 17 than it is to 23.
Primes that are the average of the previous prime and the next prime are called balanced primes. Sphenic numbers that are the product of three balanced primes are not numerous and there are none in the range up to 40,000. However, in the range up to one million, there are 82 of them. They are not listed in the OEIS and they are as follows (permalink):
41605, 45845, 55915, 68105, 69695, 98845, 135805, 149195, 157145, 160855, 165635, 173045, 182515, 194245, 201745, 206455, 222305, 227495, 250955, 258905, 271135, 277465, 292295, 292805, 297595, 314555, 322645, 324095, 337955, 362255, 393515, 400415, 441955, 462955, 464545, 465505, 476495, 479305, 486995, 490495, 505355, 512605, 512945, 525055, 564845, 575405, 593965, 606055, 625615, 634045, 640385, 640505, 688915, 709405, 723455, 740345, 743395, 762005, 766945, 769295, 773315, 779795, 779995, 785195, 798205, 819155, 839105, 845105, 858695, 865855, 876355, 877945, 881555, 931795, 941905, 954095, 960055, 963805, 963895, 971395, 989245, 999085
Take 41605 = 5 x 53 x 157 as an example. We have 5 as the average of 3 and 7, 53 as the average of 47 and 59 and 157 as the average of 151 and 163,
What about sphenic numbers that are the products of Sophie Germain primes? These turn out to be more numerous. There are 655 in the range up to 40000 out of a total of 7720. These numbers form OEIS A157346:
A157346 | Products of 3 distinct Sophie Germain primes. |
The initial members are (permalink):
30, 66, 110, 138, 165, 174, 230, 246, 290, 318, 345, 410, 435, 498, 506, 530, 534, 615, 638, 678, 759, 786, 795, 830, 890, 902, 957, 1038, 1074, 1130, 1146, 1166, 1245, 1265, 1310, 1334, 1335, 1353, 1398, 1434, 1506, 1595, 1686, 1695, 1730, 1749, 1758, 1790
Take 30 = 2 x 3 x 5 as an example. 2 x 2 + 1 = 5, 2 x 3 + 1 = 7 and 2 x 5 + 1 = 11.
What about sphenic numbers that are the product of primes that are the smaller of twin prime pairs? There are 277 such numbers in the range up to 40,000 and they are not listed in the OEIS. These numbers are as follows (permalink):
165, 255, 435, 561, 615, 885, 935, 957, 1065, 1353, 1479, 1515, 1595, 1605, 1947, 2055, 2091, 2235, 2255, 2343, 2465, 2685, 2865, 2955, 3009, 3245, 3333, 3405, 3485, 3531, 3567, 3585, 3621, 3905, 4035, 4215, 4521, 4665, 4917, 5015, 5133, 5151, 5205, 5423, 5457, 5555, 5885, 5907, 5945, 6035, 6177, 6285, 6303, 6465, 6501, 6915, 6987, 7257, 7491, 7535, 7599, 7667, 7815, 7887, 8195, 8535, 8555, 8585, 8733, 8787, 8877, 8985, 9095, 9129, 9255, 9273, 9309, 9615, 9741, 9845, 9885, 10047, 10263, 10295, 10505, 10835, 11033, 11451, 11577, 11645, 11919, 12095, 12135, 12189, 12315, 12405, 12423, 12485, 12567, 12665, 12855, 12963, 13079, 13145, 13161, 13215, 13277, 13719, 13827, 14223, 14331, 14555, 14645, 14795, 15213, 15215, 15285, 15455, 15465, 15515, 15573, 15735, 15861, 15915, 16235, 16365, 16617, 16745, 16851, 17105, 17139, 17193, 17265, 17697, 17877, 18327, 18435, 18777, 18821, 18887, 18939, 19085, 19155, 19295, 19335, 19515, 19749, 19767, 19785, 19865, 20009, 20213, 20315, 20361, 20705, 20793, 20945, 21153, 21369, 21405, 21513, 21605, 21747, 21765, 21935, 21981, 22017, 22215, 22305, 22649, 22791, 22865, 23045, 23403, 23493, 23511, 23705, 23885, 24105, 24231, 24249, 24285, 24447, 25005, 25355, 25455, 25619, 25815, 25955, 26373, 26435, 26571, 26609, 26697, 26805, 27057, 27093, 27291, 27695, 27863, 27921, 28065, 28085, 28155, 28281, 28565, 28655, 28965, 29019, 29073, 29087, 29181, 29235, 29397, 29495, 29795, 29955, 30189, 30405, 30545, 30549, 31215, 31295, 31305, 31467, 31565, 31665, 31683, 31737, 31935, 32021, 32115, 32219, 32421, 32691, 32915, 32945, 33087, 33473, 33555, 33609, 33627, 33807, 33935, 34005, 34023, 34133, 34563, 34617, 34635, 34655, 34869, 35003, 35013, 35085, 35255, 35615, 35715, 35717, 35855, 36003, 36245, 36453, 36635, 36695, 36839, 37497, 37983, 37985, 38127, 38235, 38253, 38865, 39005, 39155, 39185, 39855
Take 165 = 3 x 5 x 11 as an example. 3 is the smaller of the twin prime pair (3, 5), 5 is the smaller of the twin prime pair (5, 7) and 11 is the smaller of the twin prime pair (11, 13). At first sight, there appears to be too many numbers because the lesser prime of a twin prime pair is always a strong prime if the prime is greater than 5. This is because in a twin prime pair \( (p, p + 2)\) with \(p > 5\), \(p\) is always a strong prime, since \(3\) must divide \(p − 2\), which cannot be prime.
Once we take out any sphenic numbers containing 3 or 5, we are left with only 21 sphenic numbers that contain primes that are both strong and the lesser of twin primes. These numbers are (permalink):
5423, 7667, 11033, 13079, 13277, 18821, 18887, 20009, 20213, 22649, 25619, 26609, 27863, 29087, 32021, 32219, 33473, 34133, 35003, 35717, 36839
Take 5423 = 11 x 17 x 29 as an example. This example was given earlier as the first member of OEIS A363782 where we saw that 11 is closer to 13 than it is to 7, 17 is closer to 19 than it is to 13 and 29 is closer to 31 than it is to 23. Now, additionally, it can be seen that 11 is the lesser of (11, 13), 17 is the lesser of (17, 19) and 29 is the lesser of (29, 31).
What about sphenic numbers that are the product of primes that are the larger of twin prime pairs? There are 166 in the range up to 40,000 and they are (permalink):
455, 665, 1085, 1235, 1505, 1729, 2015, 2135, 2555, 2795, 2821, 2945, 3605, 3815, 3913, 3965, 4085, 4123, 4745, 4865, 5285, 5551, 5719, 5795, 6335, 6643, 6665, 6695, 6755, 6935, 6965, 7085, 7657, 8015, 8113, 8435, 9035, 9331, 9373, 9455, 9485, 9709, 9785, 9815, 9905, 9919, 10355, 10621, 10955, 11315, 11765, 12215, 12545, 12649, 12935, 13115, 13205, 13237, 13699, 13741, 14345, 14497, 14735, 14885, 15067, 15155, 15665, 15695, 15841, 15965, 16205, 16471, 16895, 17195, 17329, 17563, 17615, 18031, 18109, 18305, 18335, 18361, 18395, 18487, 18905, 19985, 20083, 20345, 20839, 21035, 21545, 21665, 21755, 21931, 21973, 22145, 22265, 22351, 22505, 22685, 22895, 23135, 23405, 23435, 23653, 24073, 24583, 24661, 25327, 25441, 25669, 25745, 25753, 26467, 26885, 26923, 27365, 28055, 28145, 28385, 28483, 28805, 29015, 29419, 29735, 29885, 29915, 30065, 30095, 30163, 30457, 30845, 30905, 31003, 31171, 31415, 31759, 32053, 32465, 32767, 32809, 33155, 33245, 33995, 34099, 34333, 35495, 35735, 35929, 36043, 36155, 36785, 37115, 37205, 37297, 37355, 37595, 37639, 38255, 38311, 38915, 39065, 39277, 39403, 39785, 39995
Take 455 = 5 x 7 x 13 as an example. 5 is the larger of the twin prime pair (3, 5), 7 is the larger of the twin prime pair (5, 7) and 13 is the larger of the twin prime pair (11, 13).
Clearly we could go on and on but that's probably enough for now. The same approach could be applied to composite numbers that are the product of two distinct prime, four distinct primes etc. For example, numbers that are the product of four distinct strong primes form OEIS A363167:
A363167 | Products of four distinct strong primes. |
The 47 members up to one million are (permalink):
200651, 222343, 283679, 319957, 363341, 385033, 408221, 428417, 452353, 463573, 483923, 491249, 513689, 526031, 544357, 546601, 547723, 580261, 605693, 671143, 688721, 696377, 698819, 739211, 740333, 742951, 743699, 747881, 771661, 774367, 783343, 790801, 808027, 820369, 838013, 871607, 876293, 878713, 883949, 889559, 928609, 932437, 947903, 970717, 973709, 984533, 989791
Take 200651 = 11 x 17 x 29 x 37 as an example. 11 is closer to 13 than it is to 7, 17 is closer to 19 than it is to 13, 29 is closer to 31 than it is to 23 and 37 is closer to 41 than it is to 31. All the properties mentioned in this post are independent of the number base used.
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