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EXAMPLE
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11 is here: 2^11 has 11 nonprime divisors; c(11)=A055079(11) could not have r=2, 3, 4 or more distinct prime divisors because 11+{2, 3, 4, 5, 6, 7, 8, 9, ...} values of corresponding d[c(11)]={13, 14, 15, ..} had 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2 non-distinct prime divisors which provides upper bound for r... in contradiction with demanded values: 2, 3, 4, 5, 6, 7, ... This is why A055079(11)=2048. Larger cases are handled in a similar way.
The largest known value of A055079 is A055079(15239)=2^15239 which has 4588 decimal digits.
A protocol for 15239 is as follows: u=15239; t0=Table[s, {s, 0, 17}]; t1=Table[mr[w], {w, u, u+17}]; t2=t1-t0; g=Table[{w, mr[w]}, {w, u, u+17}]; i1=TimeUsed[]; Write["a(bad)tx1", u, t1, t2, g]; 15239
Supposed number of A001221(x) which should be larger or equal than A001222[d(x)]: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}
A001222[d(x)] {3, 6, 1, 2, 2, 4, 2, 6, 2, 5, 4, 5, 2, 5, 2, 3, 5, 4}
A001222[d(x)] - A001221(x) (negative value means "nasty case") {3, 5, -1, -1, -2, -1, -4, -1, -6, -4, -6, -6, -10, -8, -12, -12, -11, -13} numbers (corresponding d(x) values for some x) together with A001222[d(x)] {{15239, 3}, {15240, 6}, {15241, 1}, {15242, 2}, {15243, 2}, {15244, 4}, {15245, 2}, {15246, 6}, {15247, 2}, {15248, 5}, {15249, 4}, {15250, 5}, {15251, 2}, {15252, 5}, {15253, 2}, {15254, 3}, {15255, 5}, {15256, 4}}
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