More on A + B + C = D and A + B = C

In my previous post titled A + B + C = D, I generated a list of 44 "d" numbers with the property that:$$a+b+c=d$$where \(a\), \(b\), \(c\) and \(d\) share the same digits and \(a<b<c\). The list of such numbers, up to 40000, is:

4617, 4851, 5103, 5184, 5913, 6021, 6129, 6192, 6219, 6291, 6921, 7182, 7281, 7416, 7614, 8145, 8154, 8253, 8325, 8451, 8514, 8523, 8541, 9135, 9216, 9234, 9324, 9612, 9621, 31860, 31905, 36171, 36711, 37116, 37161, 38061, 38106, 38151, 38214, 38511, 39051, 39105, 39411, 39501

In my post I noted the significant gap between sequence member 9621 and sequence member 31860. This was unfortunate as I'm currently 27811 days old and so all the numbers associated with my diurnal age, now and for some considerable time into the future, fall into this gap. The problem arises because I'm only displaying the sums, \(d\), resulting from the addition of the three smaller numbers \(a\), \(b\) and \(c\). I've gone back and rectified this.

What I needed to do was to extend the range of \(d\) numbers to 100,000 and then include all \(a\), \(b\), \(c\) and \(d\) numbers in a range let's say from 27810 to 40000. Doing this I get a far more useful list of numbers. Here is the expanded list (Google Doc link):

27810, 27864, 27891, 27936, 27954, 27963, 28017, 28026, 28035, 28062, 28071, 28116, 28125, 28161, 28170, 28179, 28197, 28206, 28215, 28260, 28269, 28359, 28413, 28458, 28467, 28476, 28512, 28521, 28539, 28548, 28593, 28611, 28647, 28674, 28692, 28701, 28710, 28719, 28746, 28764, 28791, 28845, 28854, 28863, 28917, 28935, 28953, 28962, 28971, 29016, 29034, 29043, 29061, 29106, 29160, 29178, 29187, 29268, 29304, 29340, 29358, 29367, 29385, 29394, 29439, 29448, 29475, 29493, 29538, 29583, 29601, 29610, 29628, 29637, 29673, 29682, 29718, 29754, 29763, 29781, 29817, 29835, 29853, 29871, 29961, 30015, 30150, 30159, 30168, 30195, 30285, 30294, 30429, 30492, 30519, 30582, 30591, 30627, 30681, 30726, 30825, 30852, 30924, 30942, 30951, 31059, 31068, 31149, 31158, 31176, 31185, 31464, 31491, 31509, 31590, 31599, 31608, 31635, 31644, 31653, 31680, 31689, 31698, 31761, 31788, 31806, 31815, 31842, 31860, 31869, 31878, 31896, 31905, 31959, 31968, 31986, 31995, 32049, 32076, 32085, 32148, 32418, 32481, 32490, 32499, 32580, 32607, 32679, 32697, 32760, 32769, 32796, 32814, 32841, 32850, 32859, 32886, 32895, 32904, 32958, 32967, 32976, 32985, 32994, 34029, 34119, 34128, 34164, 34182, 34218, 34281, 34299, 34461, 34614, 34641, 34812, 34821, 34911, 34992, 35019, 35082, 35091, 35109, 35118, 35190, 35217, 35271, 35631, 35712, 35721, 35802, 35820, 35829, 35892, 35910, 35982, 35991, 36018, 36108, 36117, 36135, 36144, 36153, 36171, 36198, 36261, 36279, 36288, 36297, 36315, 36351, 36414, 36513, 36531, 36621, 36711, 36729, 36792, 36810, 36819, 36918, 36927, 36972, 36981, 37116, 37125, 37161, 37179, 37197, 37215, 37251, 37269, 37296, 37521, 37611, 37629, 37719, 37917, 37962, 38016, 38061, 38106, 38115, 38142, 38151, 38160, 38169, 38187, 38196, 38214, 38241, 38286, 38295, 38412, 38511, 38529, 38592, 38610, 38619, 38682, 38691, 38817, 38826, 38871, 38916, 38925, 38952, 38961, 39015, 39024, 39042, 39051, 39105, 39150, 39159, 39177, 39186, 39195, 39204, 39258, 39285, 39402, 39411, 39420, 39492, 39501, 39510, 39528, 39582, 39591, 39618, 39627, 39672, 39681, 39717, 39726, 39762, 39816, 39825, 39852, 39861, 39942, 39951

Once a number is identified (whether it be an \(a\), \(b\), \(c\) or \(d\) number), then the other three members of the quadruplet can be called up. For example, \(27864\) is the \(b\) with \(a=27648, c=28764\) and \(d = 84276\). such that:$$27648+27864+28764=84276$$For want of a better term we might call these sorts of numbers \(a,b,c,d\) numbers. I have the following entry for them in my Bespoken for Sequences document on Google Docs where code can be located as well by following this permalink but execution may time out in SageMathCell so Jupyter notebook may be required or any software capable of running Python code:

Bespoken for Sequences link

A similar thing can be done with A + B = C. Here is the list of numbers between 27809 and 40000 with the property described below in Bespoken for Sequences where code can be located as well as by following this permalink but execution may time out in SageMathCell and so Jupyter notebook may be required or any software capable of running Python code:

Bespoken for Sequences link

27846, 27936, 27945, 27954, 27963, 28035, 28503, 28530, 28539, 28746, 28935, 28953, 29016, 29106, 29160, 29286, 29367, 29376, 29385, 29457, 29502, 29574, 29601, 29637, 29664, 29691, 29736, 29754, 29763, 29853, 30267, 30276, 30285, 30465, 30627, 30654, 30762, 30825, 31698, 31905, 31950, 31968, 32697, 32706, 32760, 32769, 32796, 32850, 32895, 32967, 32976, 32985, 34128, 34182, 34218, 34281, 34497, 34569, 34578, 34587, 34659, 34749, 34758, 34785, 34812, 34821, 34857, 34875, 34947, 35082, 35289, 35703, 35730, 35784, 35874, 35901, 35910, 36198, 36297, 36918, 36972, 37296, 37449, 37503, 37584, 37854, 38124, 38142, 38214, 38241, 38412, 38421, 38529, 38574, 38754, 38925, 38952, 39105, 39150, 39285, 39852

For example, 27846 is a member of this sequence with \(a=27846, b=46782\) and \(c=74628\):$$27846 + 46782 = 74628$$The code and details of all such triplets can be found in the links provided.