Gapful Numbers

 Numbers Aplenty has this to say about gapful numbers:

L. Colucci calls a number \(n\) of at least 3 digits a gapful number if \(n\)  is divisible by the number formed by the first and last digit of \(n\). For example, 583 is gapful because it is divisible by 53. About the 7.46% of the numbers are gapful.

The smallest Pythagorean triple made of gapful numbers is (120, 160, 200) and the first such primitive triple is (3159, 29440, 29609).

The concept can be extended to other bases. For example, the number 240 is gapful in all the bases from 2 to 10 and the number 54288 is gapful in all the bases from 2 to 16.

The first gapful numbers are 100, 105, 108, 110, 120, 121, 130, 132, 135, 140, 143, 150, 154, 160, 165, 170, 176, 180, 187, 190, 192, 195, 198, 200, 220, 225, 231, 240, 242, 253  

Figure 1 shows the representation of \(54288 = 2^4 \times 3^2 \times 13 \times 29\) in bases from 2 to 16:

Figure 1

Looking at the first and last digits in these different representations we see that:

• Base 2 --> 10 which is 2

• Base 3 --> 20 which is 6

• Base 4 --> 30 which is 12

• Base 5 --> 33 which is 18

• Base 6 --> 10 which is 6

• Base 7 --> 33 which is 24

• Base 8 --> 10 which is 8

• Base 9 --> 80 which is 72

• Base 10 --> 58 which is 58

• Base 11 --> 33 which is 36

• Base 12 --> 20 which is 24

• Base 13 --> 10 which is 13

• Base 14 --> 1a which is 24

• Base 15 --> 13 which is 18

• Base 16 --> d0 which is 13 x 16 = 208

All these numbers (2, 6, 8, 12 etc.) are divisors of 54288 and this why the number is so special.

One way to make these gapful numbers less numerous is to impose an additional criterion. For example, let's require that the sum of the number's digits is equal to the concatenation of the first and last digits. In the range between 1 and 40000, this reduces the numbers satisfying all criteria to 359. The numbers are (permalink):

190, 192, 195, 198, 1090, 1092, 1095, 1098, 1180, 1183, 1185, 1188, 1270, 1272, 1274, 1275, 1278, 1360, 1365, 1368, 1450, 1452, 1455, 1456, 1458, 1540, 1545, 1547, 1548, 1630, 1632, 1635, 1638, 1720, 1725, 1728, 1729, 1810, 1812, 1815, 1818, 1900, 1904, 1905, 1908, 2992, 2997, 10090, 10092, 10094, 10095, 10096, 10098, 10180, 10185, 10188, 10270, 10272, 10275, 10278, 10279, 10360, 10365, 10368, 10450, 10452, 10455, 10458, 10540, 10543, 10545, 10548, 10630, 10632, 10635, 10638, 10720, 10724, 10725, 10727, 10728, 10810, 10812, 10815, 10816, 10818, 10900, 10901, 10905, 10908, 11080, 11085, 11088, 11170, 11172, 11175, 11178, 11260, 11265, 11268, 11350, 11352, 11354, 11355, 11358, 11440, 11445, 11448, 11530, 11532, 11535, 11536, 11538, 11620, 11625, 11628, 11710, 11712, 11713, 11715, 11718, 11800, 11805, 11808, 12070, 12072, 12075, 12078, 12160, 12165, 12168, 12250, 12252, 12255, 12256, 12257, 12258, 12340, 12345, 12348, 12430, 12432, 12435, 12438, 12520, 12525, 12528, 12610, 12612, 12614, 12615, 12618, 12700, 12705, 12708, 13060, 13065, 13068, 13150, 13152, 13155, 13158, 13240, 13244, 13245, 13248, 13330, 13332, 13335, 13338, 13420, 13425, 13428, 13510, 13512, 13515, 13518, 13600, 13605, 13608, 14050, 14052, 14053, 14055, 14058, 14140, 14145, 14148, 14230, 14232, 14235, 14238, 14320, 14325, 14328, 14410, 14412, 14415, 14416, 14418, 14500, 14504, 14505, 14508, 15040, 15045, 15048, 15130, 15132, 15134, 15135, 15136, 15138, 15220, 15223, 15225, 15228, 15310, 15312, 15315, 15317, 15318, 15400, 15405, 15408, 15409, 16030, 16032, 16035, 16038, 16120, 16125, 16128, 16210, 16212, 16215, 16218, 16300, 16305, 16308, 17020, 17024, 17025, 17028, 17110, 17112, 17115, 17118, 17119, 17200, 17205, 17208, 18010, 18012, 18015, 18016, 18018, 18100, 18105, 18108, 19000, 19005, 19008, 21897, 21980, 21984, 22792, 22793, 22880, 22975, 22977, 23780, 23782, 23784, 23787, 23875, 23960, 23961, 23968, 24591, 24596, 24597, 24680, 24772, 24775, 24860, 24863, 24864, 24867, 25580, 25584, 25675, 25677, 25760, 25762, 25766, 25851, 25940, 25944, 25947, 26480, 26481, 26487, 26488, 26575, 26660, 26664, 26752, 26757, 26840, 26933, 26936, 27297, 27380, 27384, 27475, 27560, 27567, 27740, 27741, 27742, 27744, 27748, 27837, 27920, 27925, 28280, 28371, 28375, 28377, 28460, 28464, 28640, 28647, 28732, 28739, 28820, 28824, 28825, 28917, 29180, 29184, 29187, 29275, 29276, 29360, 29457, 29540, 29544, 29631, 29720, 29722, 29725, 29727, 29900, 29904, 39990, 39996, 39997

For example, 39997 has a first and last digit concatenation of 37 with a digit of 37 and a factoriation of 23 x 37 x 47. 

If instead, we require the product of the number's digit instead then only five numbers satisfy: 135, 11232, 12132, 12312, 13212 (permalink). For example, 13212 = 2^2 x 3^2 x 367 has a digit product of 12 which is a divisor of the number and 12 is also the concatenation of the first and last digits of the number.

Another criterion that could be applied is to require that the first and last digits be the same. In this case 227 numbers satisfy in the range up to 40000. They are (permalink):

121, 242, 363, 484, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3333, 3663, 3993, 4004, 4224, 4444, 4664, 4884, 5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 6006, 6336, 6666, 6996, 7007, 7777, 8008, 8448, 8888, 9009, 9999, 10021, 10131, 10241, 10351, 10461, 10571, 10681, 10791, 10901, 11011, 11121, 11231, 11341, 11451, 11561, 11671, 11781, 11891, 12001, 12111, 12221, 12331, 12441, 12551, 12661, 12771, 12881, 12991, 13101, 13211, 13321, 13431, 13541, 13651, 13761, 13871, 13981, 14091, 14201, 14311, 14421, 14531, 14641, 14751, 14861, 14971, 15081, 15191, 15301, 15411, 15521, 15631, 15741, 15851, 15961, 16071, 16181, 16291, 16401, 16511, 16621, 16731, 16841, 16951, 17061, 17171, 17281, 17391, 17501, 17611, 17721, 17831, 17941, 18051, 18161, 18271, 18381, 18491, 18601, 18711, 18821, 18931, 19041, 19151, 19261, 19371, 19481, 19591, 19701, 19811, 19921, 20042, 20152, 20262, 20372, 20482, 20592, 20702, 20812, 20922, 21032, 21142, 21252, 21362, 21472, 21582, 21692, 21802, 21912, 22022, 22132, 22242, 22352, 22462, 22572, 22682, 22792, 22902, 23012, 23122, 23232, 23342, 23452, 23562, 23672, 23782, 23892, 24002, 24112, 24222, 24332, 24442, 24552, 24662, 24772, 24882, 24992, 25102, 25212, 25322, 25432, 25542, 25652, 25762, 25872, 25982, 26092, 26202, 26312, 26422, 26532, 26642, 26752, 26862, 26972, 27082, 27192, 27302, 27412, 27522, 27632, 27742, 27852, 27962, 28072, 28182, 28292, 28402, 28512, 28622, 28732, 28842, 28952, 29062, 29172, 29282, 29392, 29502, 29612, 29722, 29832, 29942, 30063, 30393, 30723, 31053, 31383, 31713, 32043, 32373, 32703, 33033, 33363, 33693, 34023, 34353, 34683, 35013, 35343, 35673, 36003, 36333, 36663, 36993, 37323, 37653, 37983, 38313, 38643, 38973, 39303, 39633, 39963

For example, 39963 = 3 x 7 x 11 x 173 has first and last digits the same and 33 divides the number.