Smarandache Symmetric Sequence

The Smarandache symmetric sequence is formed by the decimal concatenation of the blocks 1, 2, ..., , , ..., 1 for , with the decreasing part empty for . Here and are the ceiling function and floor function. The terms begin 1, 11, 121, 1221, 12321, 123321, ... (OEIS A007907) (Smarandache 1993, p. 19). The symmetry is between integer blocks, so terms containing multidigit blocks need not be palindromic numbers.

The odd-index terms concatenate 1, 2, ..., , , ..., 1 and begin 1, 121, 12321, 1234321, ... (OEIS A173426). The even-index terms repeat the central block and begin 11, 1221, 123321, 12344321, ... (OEIS A259937). The smallest prime among the odd-index terms is . A 17350-digit probable prime is reported at (OEIS A173426) (Numberphile 2021). The index list OEIS A359148 includes a placeholder initial 1 before 10 and 2446; this 1 does not identify a prime, since . Whether infinitely many terms are prime is unknown.

For an integer base , let be the value obtained by concatenating the base- representations of 1, 2, ..., , , ..., 1 for . The following table separates digit strings recorded as decimal integers from their values in the original base. The value sequence in base 3 also has a zeroth term 0.

For example, and are primes. The decimal integer 121021 is also prime, although it is a different value.

If , then , where is the repunit of length in base . Thus , and every with is composite. The first possible prime therefore occurs at . Writing gives the candidate

Indeed, the ascending and descending blocks flanking have values and , respectively. The two-digit central block therefore gives , which simplifies to the formula above. The prime values among these candidates begin 13, 439, 27961, 522134761, ... (OEIS A260852), corresponding to bases 2, 3, 4, 6, ... (OEIS A260343). In particular, the base-4 example is .