I have modified my original search to now simply count the number of solutions. My program first created a list of 3 element subsets of the set of 21 primes starting at 7 for which their total is prime. This found 618 such subsets. I then systematically search all of the ways to choose 7 of these subsets such that they form a disjoint partition of the original set and where the total of the reverse of the sub-totals is equal to the total of all the original primes. There are 6,602,055,918,461,784 ways of choosing 7 sets from the 618 so to speed things, each time I pick a new set from the 618 I make sure it does not have any elements in common with the previous sets picked and I also stop the loop when the running total exceeds to original total. Still this code is going to take quite some time, perhaps several days as I am not able to let it run continuously and must make use of state saves to stop and continue the search at various times. I am pasting my search code below so if somebody has a fast computer to try running it on perhaps you can get a grand total faster than I can.
prms = Table[Prime[t], {t, 4, 24}];
RevD[x_] := FromDigits[Reverse[IntegerDigits[Total[x]]]];
IsDisj[x_] :=
If[Length[Flatten[x]] == Length[Union[Flatten[x]]], Return[True],
Return[False]];
tot = Total[prms];
cnt = 0;
tups = {};
subs = Subsets[prms, {3}];
For[i = 1, i <= Length[subs], ++i,
If[PrimeQ[Total[subs[[i]]]],
cnt++;
tups = Append[tups, subs[[i]]];
];
];
ans = 316;
For[i1 = 7, i1 <= cnt - 6, ++i1,
Print[i1, "/", cnt - 6, " solutions: ", ans];
set1 = tups[[i1]];
v1 = RevD[set1];
For[i2 = i1 + 1, i2 <= cnt - 5, ++i2,
set2 = tups[[i2]];
v2 = RevD[set2];
If[IsDisj[{set1, set2}] && v1 + v2 + 13*5 < tot,
For[i3 = i2 + 1, i3 <= cnt - 4, ++i3,
set3 = tups[[i3]];
v3 = RevD[set3];
If[IsDisj[{set1, set2, set3}] && v1 + v2 + v3 + 13*4 < tot,
For[i4 = i3 + 1, i4 <= cnt - 3, ++i4,
set4 = tups[[i4]];
v4 = RevD[set4];
If[
IsDisj[{set1, set2, set3, set4}] &&
v1 + v2 + v3 + v4 + 13*3 < tot,
For[i5 = i4 + 1, i5 <= cnt - 2, ++i5,
set5 = tups[[i5]];
v5 = RevD[set5];
If[IsDisj[{set1, set2, set3, set4, set5}] &&
v1 + v2 + v3 + v4 + v5 + 13*2 < tot,
For[i6 = i5 + 1, i6 <= cnt - 1, ++i6,
set6 = tups[[i6]];
v6 = RevD[set6];
If[IsDisj[{set1, set2, set3, set4, set5, set6}] &&
v1 + v2 + v3 + v4 + v5 + v6 + 13 <= tot,
For[i7 = i6 + 1, i7 <= cnt, ++i7,
set7 = tups[[i7]];
v7 = RevD[set7];
If[IsDisj[{set1, set2, set3, set4, set5, set6,
set7}] &&
v1 + v2 + v3 + v4 + v5 + v6 + v7 == Total[prms],
ans++;
];];];];];];];];];];];];];
update
