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.

%I A369112 #13 May 21 2024 11:29:34
%S A369112 -1,3,4,6,8,12,-1,24,30,32,-1,48,60,-1,72,224,168,128,-1,180,192,390,
%T A369112 -1,240,-1,972,360,384,420,432,480,-1,-1,720,1280,840,864,-1,2112,-1,
%U A369112 1512,1152,1260,-1,1440,3584,3120,-1,2700,2688,-1,3024,-1,2310,3780,2592
%N 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.
%C A369112 The n-th prime signature is the prime signature of A025487(n).
%C A369112 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.
%C A369112 a(n) = -1 if A025487(n) is a number of either
%C A369112   (1) the form b^e, where b > 0 is a nonprime and e > 1, or
%C A369112   (2) the form 2^p, where p is a prime but 2^p - 1 is composite.
%C A369112 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.
%F A369112 a(n) = A025487(n) iff A025487(n) - 1 is prime.
%e A369112 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.
%e A369112 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.
%e A369112 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.
%e A369112 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.
%e A369112 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).
%e A369112 .
%e A369112                               |   numbers k of form    |
%e A369112   n | form  | numbers of form | such that k-1 is prime | a(n)
%e A369112   --+-------+-----------------+------------------------+-----
%e A369112   1 |   1   | 1               | (none exist)           |  -1
%e A369112   2 |   p   | 2, 3, 5, 7, ... | 3                      |   3
%e A369112   3 |  p^2  | 4, 9, 25, ...   | 4                      |   4
%e A369112   4 |  p*q  | 6, 10, 14, ...  | 6, 14, 38, 62, 74, ... |   6
%e A369112   5 |  p^3  | 8, 27, 125, ... | 8                      |   8
%e A369112   6 | p^2*q | 12, 18, 20, ... | 12, 20, 44, 68, ...    |  12
%e A369112   7 |  p^4  | 16, 81, ...     | (none exist)           |  -1
%e A369112   8 | p^3*q | 24, 40, 54, ... | 24, 54, 104, 152, ...  |  24
%e A369112   9 | p*q*r | 30, 42, 66, ... | 30, 42, 102, 110, ...  |  30
%Y A369112 Cf. A025487, A124832.
%K A369112 sign
%O A369112 1,2
%A A369112 _Jon E. Schoenfield_, Jan 13 2024