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Partial sums of self-describing primes, where the digits are described in any order, whereas in A047841 they must be described in increasing order. The subsequence of prime partial sums of self-describing primes begins: 10153331, 75460003. What is the smallest value in the subsubsequence of self-describing prime partial sums of self-describing primes?

A digit count numerically summarizes the frequency of digits 0 through 9 in that order when they occur in a number.

This uses a different method from A108810. Here the digits are described in increasing order, whereas in A108810 they can be described in any order.

This sequence is finite, since T(x) < x for every x with at least 22 digits. Last term is a(109) = 101112213141516171819. - Schimke

A character in the Verghese (2009) novel declares that 10213223 "is the only number that describes itself when you read it." - Alonso del Arte, Jan 26 2014

From Robert G. Wilson v, May 05 2012: (Start)

If abcd... with a, b, c & d integers, then so is cdab... . As an example, since 10123133 is a term so must be 10123331, 10311233, 10313312, 10331231, 10333112, 12103133, 12103331, 12311033, 12313310, 12331031, 12333110, 31101233, 31103312, 31121033, 31123310, 31331012, 31331210, 33101231, 33103112, 33121031, 33123110, 33311012, 33311210.

Therefore 10123133 can be said to be the progenerator or the primitive self-describing number.

Also if we index the number abcd... from left to right, the sum of the odd indexes must equal the number of digits for unique even-indexed digits.

Number of terms < 10^2n: 1, 2, 6, 1043, 5498, ..., .

This sequence is finite with the last term is probably 9998979595959595848484848484848476737373737373736262626262625151515110.

(End)

This is to A001358 as A108810 is to A000040.