A373756 Table read by antidiagonals: T(n,k) is the smallest m > 1 such that m^2 - 1 and m^2 + 1 have 2n and 2k divisors, respectively, or -1 if no such m exists.

2, 4, -1, 10, 3, -1, 14, 8, 18, -1, 28560, 5, 168, 72, -1, 26, 9, 32, 360, 16068, -1, 25071688922457240, 15, 7, 68, 369465818568, 1620, -1, 56, 728, 332, 28398240, 182, 744768, 1407318, -1, 170, 11, 161245807967271241368, 98, 248872305817685706212070112080, 132, 4175536688568, 642, -1

Offset: 1

Author

Jon E. Schoenfield, Jun 16 2024

Keywords

• sign
• tabl

Comments

m=1 is excluded because m^2 - 1 would be 0.

For all m > 1, both m^2 - 1 and m^2 + 1 are nonsquares, so each has an even number of divisors.

For k=1, m^2 + 1 is a prime, so T(n,1) == 0 (mod 2) for all n.

For n=1, m^2 - 1 = (m-1)*(m+1) is a prime, which occurs only at m=2; 2^2 + 1 = 5 is also a prime, so T(1,1) = 2 and T(1,k) = -1 for k > 1.

For n=2, m^2 - 1 = (m-1)*(m+1) has 4 divisors, so (except for T(2,2) = 3) T(2,k) is the average of a twin prime pair (A014574).

Is T(n,k) > 0 for all n > 1?

Examples

			T(5,1) is the smallest integer m > 1 such that m^2 - 1 and m^2 + 1 have 10 and 2 divisors, respectively; since m^2 - 1 cannot be the 9th power of a prime, this requires that p^4 * q + 1 = m^2 = r - 1, where p, q, and r are distinct primes. The smallest such m is 28560, which gives a solution with p = 13, q = 28559, r = 815673601.
T(5,5) is the smallest integer m > 1 such that m^2 - 1 and m^2 + 1 each have 10 divisors; since neither m^2 - 1 nor m^2 + 1 can be the 9th power of a prime, this is the smallest m such that p^4 * q + 1 = m^2 = r^4 * s - 1, where p, q, r, and s are distinct primes: 22335421^4 * 248872305817685706212070112079 + 1 = 248872305817685706212070112080^2 = 13^4 * 2168601400616633822685176617536070987718973054081571441 - 1.
The first eight antidiagonals of the table are shown below.
.
  n\k|                 1   2   3        4            5      6       7  8
  ---+------------------------------------------------------------------
   1 |                 2  -1  -1       -1           -1     -1      -1 -1
   2 |                 4   3  18       72        16068   1620 1407318
   3 |                10   8 168      360 369465818568 744768
   4 |                14   5  32       68          182
   5 |             28560   9   7 28398240
   6 |                26  15 332
   7 | 25071688922457240 728
   8 |                56
		

Crossrefs

Cf. A000005, A002522, A005563, A014574, A069062, A193432, A347191.

Formula

Define f(m) = tau(m^2 - 1) and g(m) = tau(m^2 + 1), where tau is the number of divisors function (A000005). Then

T(n,k) = min_{ m : f(m) = 2n and g(m) = 2k },

or -1 if no such m exists.