Energetic Numbers

I was surprised today to stumble upon a new category of number called energetic numbers. Such number are reasonably common: the 10,000th such number is 103,718. These sorts of numbers form OEIS A055480: 

A055480

Energetic numbers: numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings.

The examples are given of \(142 = 14^1 + 2^7\) and \(8833 = 88^2 + 33^2\). This property is reminiscent of d-powerful numbers but with these only the individual digits can be used. For example, 27472 can be expressed as:$$27472= 2^3 + 7^4 + 4^3 + 7^5 + 2^{13}$$My diurnal age today is 27476 and, while it is not a d-powerful number, it is an energetic number because it can be expressed as follows:$$27476=27^3+4^6+7^4+6^4$$So it is almost a d-powerful number but not quite. The initial energetic numbers are:

24, 43, 63, 89, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 132, 135, 142, 153, 175, 209, 224, 226, 262, 264, 267, 283, 284, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 568, 598, 629, 739, 794, 809, 849, 935, 994, 1000

However, the OEIS reference has a link to a text file that lists the first 10,000 such numbers. Unfortunately, the file does not list the representation of the number in terms of powers of substrings. For 27472, I had to simply experiment until I found the right combination. The text file contains the C program code used to generate the list. I used Google's Gemini to convert the code to Python but it wouldn't generate any output using SageMathCell.

The energetic numbers will contain the d-powerful numbers as a subset. The initial d-powerful numbers are:

24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994

Comparing this sequence to OEIS A055480, it can be seen that 100 is the first energetic number that is not d-powerful because:$$ \begin{align} 100 &\neq 1^2 + 0^2 + 0^2 \text{ whereas}\\ 100 &=10^2 + 0^2 \end{align}$$Similarly for the numbers from 101 to 109. Here is a list of the energetic numbers between 27476 and 40000:

27476, 27479, 27493, 27496, 27497, 27498, 27527, 27529, 27536, 27549, 27562, 27568, 27569, 27617, 27635, 27639, 27663, 27697, 27720, 27736, 27747, 27749, 27752, 27764, 27765, 27790, 27817, 27823, 27856, 27892, 27923, 27962, 27972, 27984, 28132, 28160, 28203, 28224, 28228, 28243, 28245, 28260, 28261, 28262, 28263, 28264, 28265, 28266, 28267, 28268, 28269, 28288, 28306, 28332, 28336, 28355, 28375, 28395, 28403, 28423, 28425, 28449, 28513, 28519, 28532, 28533, 28534, 28536, 28539, 28553, 28554, 28569, 28574, 28593, 28599, 28603, 28607, 28613, 28682, 28730, 28731, 28732, 28733, 28734, 28735, 28736, 28737, 28738, 28739, 28819, 28932, 28937, 28955, 29130, 29232, 29253, 29259, 29263, 29287, 29319, 29324, 29342, 29343, 29346, 29347, 29349, 29369, 29370, 29385, 29435, 29436, 29444, 29454, 29474, 29532, 29543, 29586, 29634, 29637, 29732, 29734, 29744, 29755, 29765, 29769, 29835, 29853, 29923, 29943, 29947, 29967, 30135, 30153, 30175, 30236, 30243, 30257, 30289, 30312, 30340, 30341, 30342, 30343, 30344, 30345, 30346, 30347, 30348, 30349, 30373, 30375, 30427, 30445, 30600, 30601, 30602, 30603, 30604, 30605, 30606, 30607, 30608, 30609, 30628, 30935, 30964, 31032, 31096, 31122, 31131, 31169, 31216, 31234, 31237, 31296, 31314, 31324, 31339, 31342, 31346, 31347, 31362, 31366, 31376, 31385, 31393, 31452, 31455, 31549, 31558, 31563, 31636, 31673, 31695, 31852, 31920, 31921, 31922, 31923, 31924, 31925, 31926, 31927, 31928, 31929, 31932, 32052, 32072, 32097, 32214, 32234, 32235, 32252, 32254, 32256, 32275, 32296, 32297, 32325, 32326, 32327, 32342, 32355, 32364, 32367, 32368, 32436, 32437, 32492, 32524, 32525, 32527, 32528, 32548, 32562, 32565, 32587, 32588, 32595, 32635, 32637, 32655, 32672, 32722, 32731, 32743, 32749, 32780, 32781, 32782, 32783, 32784, 32785, 32786, 32787, 32788, 32789, 32792, 32793, 32800, 32801, 32802, 32803, 32804, 32805, 32806, 32807, 32808, 32809, 32810, 32811, 32812, 32813, 32814, 32815, 32816, 32817, 32818, 32819, 32820, 32821, 32822, 32823, 32824, 32825, 32826, 32827, 32828, 32829, 32830, 32831, 32832, 32833, 32834, 32835, 32836, 32837, 32838, 32839, 32840, 32841, 32842, 32843, 32844, 32845, 32846, 32847, 32848, 32849, 32850, 32851, 32852, 32853, 32854, 32855, 32856, 32857, 32858, 32859, 32860, 32861, 32862, 32863, 32864, 32865, 32866, 32867, 32868, 32869, 32870, 32871, 32872, 32873, 32874, 32875, 32876, 32877, 32878, 32879, 32880, 32881, 32882, 32883, 32884, 32885, 32886, 32887, 32888, 32889, 32890, 32891, 32892, 32893, 32894, 32895, 32896, 32897, 32898, 32899, 32903, 32905, 32907, 32908, 32913, 32914, 32917, 32918, 32922, 32923, 32924, 32925, 32926, 32927, 32928, 32930, 32931, 32932, 32933, 32934, 32935, 32936, 32937, 32938, 32939, 32940, 32941, 32942, 32943, 32944, 32945, 32946, 32947, 32948, 32949, 32952, 32953, 32954, 32963, 32965, 32968, 32969, 32973, 32978, 32979, 32980, 32981, 32982, 32983, 32984, 32985, 32986, 32987, 32988, 32989, 32994, 32995, 32997, 32998, 33020, 33021, 33022, 33023, 33024, 33025, 33026, 33027, 33028, 33029, 33042, 33068, 33078, 33084, 33087, 33102, 33106, 33123, 33125, 33134, 33158, 33162, 33165, 33182, 33219, 33220, 33221, 33222, 33223, 33224, 33225, 33226, 33227, 33228, 33229, 33236, 33242, 33246, 33247, 33260, 33261, 33262, 33263, 33264, 33265, 33266, 33267, 33268, 33269, 33272, 33273, 33274, 33276, 33282, 33283, 33284, 33286, 33288, 33292, 33298, 33318, 33322, 33338, 33372, 33387, 33398, 33413, 33422, 33427, 33428, 33429, 33442, 33447, 33467, 33485, 33486, 33512, 33532, 33538, 33539, 33548, 33562, 33568, 33578, 33582, 33592, 33622, 33623, 33624, 33642, 33648, 33658, 33677, 33685, 33686, 33752, 33757, 33772, 33775, 33778, 33779, 33786, 33792, 33798, 33804, 33822, 33823, 33824, 33826, 33828, 33829, 33830, 33834, 33842, 33843, 33844, 33846, 33847, 33864, 33865, 33872, 33873, 33874, 33878, 33882, 33884, 33893, 33942, 33952, 33954, 33957, 33974, 33987, 33992, 34036, 34038, 34112, 34118, 34131, 34146, 34162, 34186, 34217, 34226, 34230, 34231, 34232, 34233, 34234, 34235, 34236, 34237, 34238, 34239, 34244, 34254, 34256, 34265, 34268, 34272, 34274, 34276, 34279, 34290, 34291, 34292, 34293, 34294, 34295, 34296, 34297, 34298, 34299, 34322, 34323, 34326, 34327, 34342, 34366, 34367, 34374, 34377, 34387, 34388, 34402, 34423, 34439, 34472, 34474, 34498, 34525, 34526, 34528, 34529, 34542, 34562, 34568, 34578, 34638, 34652, 34672, 34674, 34677, 34689, 34698, 34727, 34728, 34739, 34746, 34762, 34766, 34770, 34771, 34772, 34773, 34774, 34775, 34776, 34777, 34778, 34779, 34784, 34786, 34832, 34834, 34836, 34852, 34869, 34872, 34873, 34874, 34892, 34922, 34924, 34925, 34928, 34947, 34948, 34980, 34981, 34982, 34983, 34984, 34985, 34986, 34987, 34988, 34989, 35022, 35024, 35032, 35080, 35081, 35082, 35083, 35084, 35085, 35086, 35087, 35088, 35089, 35178, 35203, 35205, 35208, 35213, 35216, 35220, 35221, 35222, 35223, 35224, 35225, 35226, 35227, 35228, 35229, 35231, 35233, 35243, 35245, 35247, 35248, 35274, 35284, 35285, 35287, 35289, 35323, 35332, 35352, 35358, 35372, 35378, 35427, 35447, 35487, 35488, 35522, 35552, 35598, 35625, 35628, 35630, 35662, 35688, 35738, 35809, 35825, 35845, 35848, 35885, 35887, 35912, 35930, 35931, 35932, 35933, 35934, 35935, 35936, 35937, 35938, 35939, 35978, 35979, 35992, 35994, 36104, 36114, 36122, 36144, 36152, 36158, 36164, 36232, 36234, 36237, 36239, 36253, 36254, 36258, 36274, 36275, 36277, 36278, 36283, 36292, 36294, 36296, 36314, 36324, 36327, 36332, 36342, 36343, 36346, 36348, 36368, 36382, 36385, 36408, 36412, 36417, 36435, 36437, 36438, 36454, 36457, 36472, 36473, 36474, 36478, 36492, 36497, 36522, 36544, 36547, 36558, 36567, 36568, 36637, 36656, 36657, 36693, 36708, 36719, 36722, 36723, 36724, 36731, 36733, 36742, 36743, 36746, 36748, 36762, 36775, 36789, 36837, 36854, 36877, 36892, 36922, 36924, 36942, 36980, 36981, 36982, 36983, 36984, 36985, 36986, 36987, 36988, 36989, 37002, 37045, 37072, 37083, 37102, 37177, 37215, 37216, 37220, 37221, 37222, 37223, 37224, 37225, 37226, 37227, 37228, 37229, 37242, 37243, 37244, 37246, 37248, 37262, 37263, 37264, 37265, 37267, 37268, 37282, 37283, 37284, 37306, 37332, 37333, 37336, 37348, 37352, 37353, 37358, 37372, 37392, 37393, 37410, 37412, 37420, 37421, 37422, 37423, 37424, 37425, 37426, 37427, 37428, 37429, 37442, 37443, 37449, 37455, 37462, 37482, 37483, 37484, 37485, 37486, 37578, 37626, 37642, 37645, 37648, 37663, 37687, 37702, 37714, 37732, 37738, 37747, 37794, 37826, 37827, 37843, 37851, 37867, 37884, 37953, 37997, 38124, 38133, 38145, 38164, 38235, 38257, 38272, 38317, 38322, 38324, 38325, 38326, 38343, 38345, 38348, 38349, 38372, 38382, 38383, 38394, 38456, 38474, 38514, 38521, 38522, 38527, 38560, 38563, 38566, 38576, 38655, 38657, 38746, 38762, 38834, 38835, 38923, 38924, 39142, 39276, 39338, 39340, 39341, 39342, 39343, 39344, 39345, 39346, 39347, 39348, 39349, 39352, 39354, 39368, 39372, 39373, 39384, 39392, 39393, 39402, 39420, 39421, 39422, 39423, 39424, 39425, 39426, 39427, 39428, 39429, 39432, 39442, 39443, 39446, 39462, 39469, 39472, 39482, 39483, 39484, 39485, 39532, 39533, 39536, 39538, 39608, 39623, 39624, 39627, 39628, 39629, 39634, 39642, 39648, 39662, 39682, 39687, 39689, 39734, 39749, 39752, 39758, 39822, 39823, 39825, 39827, 39834, 39842, 39844, 39848, 39864, 39882, 39884, 39886

As with the energetic numbers 100 to 109 inclusive, runs of consecutive integers are common e.g. 30600, 30601, 30602, 30603, 30604, 30605, 30606, 30607, 30608, 30609 where we have:$$30600 = 30^3+60^2+0$$and the digits 1 to 9 can be substituted for the zero and the numbers remain energetic.