A124832 -id:A124832 - cp's OEIS frontend

A369112 a(n) is the smallest number that has the n-th prime signature and is 1 greater than a prime, or -1 if no such number exists.

-1, 3, 4, 6, 8, 12, -1, 24, 30, 32, -1, 48, 60, -1, 72, 224, 168, 128, -1, 180, 192, 390, -1, 240, -1, 972, 360, 384, 420, 432, 480, -1, -1, 720, 1280, 840, 864, -1, 2112, -1, 1512, 1152, 1260, -1, 1440, 3584, 3120, -1, 2700, 2688, -1, 3024, -1, 2310, 3780, 2592

Offset: 1

Jon E. Schoenfield, Jan 13 2024

sign

The n-th prime signature is the prime signature of A025487(n).
Among the first few hundred terms, the largest values occur where the n-th prime signature is of the form p^j * q^m, where p and q are distinct primes and j and m are coprime. E.g., a(n) = 6588344 = 2^3 * 7^7 is far larger than the previous record high, a(58) = 5120; the next record high is a(117) = 2007952544 = 2^5 * 13^7.
a(n) = -1 if A025487(n) is a number of either
  (1) the form b^e, where b > 0 is a nonprime and e > 1, or
  (2) the form 2^p, where p is a prime but 2^p - 1 is composite.
Conjecture: the converse is also true; i.e., if A025487(n) is of neither of the two forms above, then there exists at least one number that has the n-th prime signature and is 1 greater than a prime.

			a(1) = -1 because the 1st prime signature is 1 (i.e., the empty product), and 1 is the only number that has that prime signature, and 1 is not 1 greater than a prime.
a(2) = 3 because the 2nd prime signature is that of any prime p, and the only prime p such that p-1 is also a prime is 3.
a(3) = 4 because the 3rd prime signature is that of any number of the form p^2 where p is a prime, and the only such number such that p^2 - 1, which factors as (p-1)*(p+1), is a prime, is 4.
a(16) = 224 because the 16th prime signature is that of any number of the form p^5 * q where p and q are distinct primes, and the smallest such numbers are 96, 160, 224, ..., but 96 - 1 = 95 = 5*19 and 160 - 1 = 159 = 3*53, so the smallest one of those that is 1 greater than a prime is 224.
The table below gives, for n = 1..9, the form (expressed as a product of 0 or more prime powers, where p, q, and r are distinct primes) of all numbers having the n-th prime signature, along with the first few numbers of that form, the first few of those numbers (if any) that are 1 greater than a prime, and a(n).
.
                              |   numbers k of form    |
  n | form  | numbers of form | such that k-1 is prime | a(n)
  --+-------+-----------------+------------------------+-----
  1 |   1   | 1               | (none exist)           |  -1
  2 |   p   | 2, 3, 5, 7, ... | 3                      |   3
  3 |  p^2  | 4, 9, 25, ...   | 4                      |   4
  4 |  p*q  | 6, 10, 14, ...  | 6, 14, 38, 62, 74, ... |   6
  5 |  p^3  | 8, 27, 125, ... | 8                      |   8
  6 | p^2*q | 12, 18, 20, ... | 12, 20, 44, 68, ...    |  12
  7 |  p^4  | 16, 81, ...     | (none exist)           |  -1
  8 | p^3*q | 24, 40, 54, ... | 24, 54, 104, 152, ...  |  24
  9 | p*q*r | 30, 42, 66, ... | 30, 42, 102, 110, ...  |  30

Cf. A025487, A124832.

a(n) = A025487(n) iff A025487(n) - 1 is prime.

cp's OEIS Frontend

A369112 a(n) is the smallest number that has the n-th prime signature and is 1 greater than a prime, or -1 if no such number exists.

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