Equal prime and composite sums

Suppose we have a sequence of prime sums, where each term P_n is the sum of the first n primes.

P = (2), (2 + 3), (2 + 3 + 5), (2 + 3 + 5 + 7), (2 + 3 + 5 + 7 + 11), ...

P = 2, 5, 10, 17, 28, etc.

Further; suppose we have a sequence of composite sums, where each term C_m is the sum of the first m composites.

C = (4), (4 + 6), (4 + 6 + 8), (4 + 6 + 8 + 9), (4 + 6 + 8 + 9 + 10), ...

C = 4, 10, 18, 27, 37, etc.

Notice that the third term of P; P₃ (10) is equal to the second term of C; C₂ (10);

Task

Find and display the indices (n, m) and value of at least the first 6 terms of the sequence of numbers that are both the sum of the first n primes and the first m composites.

See also

OEIS:A007504 - Sum of the first n primes OEIS:A053767 - Sum of first n composite numbers OEIS:A294174 - Numbers that can be expressed both as the sum of first primes and as the sum of first composites

Raku

Let it run until I got bored and killed it. Time is total accumulated seconds since program start. <lang perl6>use Lingua::EN::Numbers:ver<2.8.2+>;

my $prime-sum = [\+] (2..*).grep: *.is-prime; my $composite-sum = [\+] (2..*).grep: !*.is-prime;

my $c-index = 0;

for ^∞ -> $p-index {

   next if $prime-sum[$p-index] < $composite-sum[$c-index];
   printf( "%20s - %11s prime sum, %12s composite sum   %5.2f seconds\n",
     $prime-sum[$p-index].&comma, ordinal-digit($p-index + 1, :u, :c),
     ordinal-digit($c-index + 1, :u, :c), now - INIT now )
     and next if $prime-sum[$p-index] == $composite-sum[$c-index];
   ++$c-index;
   redo;

};</lang>

Output:

                  10 -         3ʳᵈ prime sum,          2ⁿᵈ composite sum    0.01 seconds
               1,988 -        33ʳᵈ prime sum,         51ˢᵗ composite sum    0.01 seconds
              14,697 -        80ᵗʰ prime sum,        147ᵗʰ composite sum    0.02 seconds
              83,292 -       175ᵗʰ prime sum,        361ˢᵗ composite sum    0.03 seconds
           1,503,397 -       660ᵗʰ prime sum,      1,582ⁿᵈ composite sum    0.04 seconds
          18,859,052 -     2,143ʳᵈ prime sum,      5,699ᵗʰ composite sum    0.08 seconds
          93,952,013 -     4,556ᵗʰ prime sum,     12,821ˢᵗ composite sum    0.14 seconds
      89,171,409,882 -   118,785ᵗʰ prime sum,    403,341ˢᵗ composite sum    4.23 seconds
   9,646,383,703,961 - 1,131,142ⁿᵈ prime sum,  4,229,425ᵗʰ composite sum   76.23 seconds
 209,456,854,921,713 - 5,012,372ⁿᵈ prime sum, 19,786,181ˢᵗ composite sum  968.26 seconds
^C