Pseudo-Sphenic Number Sequences

A cubic polynomial with three real roots and rational coefficients can be written in the following form:

(a$x$ + b) $\times$  (c$x$ + d) $\times$ (e$x$ + f) 
where a, b, c, d, e and f are rational numbers

Let's modify the conditions so that a, b, c, d, e and f are integers (positive or negative) and $x$ can only take integer values greater than 1. Let's change the $x$ to an $n$ so that we have:

(a$n$ + b) $\times$ (c$n$ + d) $\times$ (e$n$ + f)

Let's take a specific example where a=1, b=0, c=1, d=1, e=2 and f=3. This gives us:

$$n \times (n+1) \times (2n+3)$$ As we plug in different values for $n$, starting with $n=2$, a series of terms arises. In this case, the terms begin:

42, 108, 220, 390, 630, 952, 1368, 1890, 2530, 3300, 4212, 5278, 6510, 7920, 9520, 11322, 13338, 15580, 18060, 20790, 23782, 27048, 30600, 34450, 38610, 43092, 47908, 53070, 58590, 64480, 70752, 77418, 84490, 91980, 99900, 108262, 117078, 126360, 136120

These terms constitute OEIS A163815 (although the terms for $n=0$ and $n=1$ are included). These sorts of sequences involve the multiplication of triple linear combinations of $n$, in this case $n$, $n+1$ and $2n+3$. This permalink leads to a SageMath algorithm that will generate a sequence of terms for varying values of a, b, c, d, e and f. 

If we impose the condition that each of the linear factors must be distinct (and this is the case for the example just shown), then we have a sequence where each member is a sort of pseudo-sphenic number that can be written as a product of three of its divisors but each divisor is a linear combination of an underlying integer. For example, take the number 27048 (my diurnal age yesterday). It can be written as:

$$27048 = 23 \times 24 \times 49\\ \text{where } 23=n, 24=n+1 \text{ and } 49=2n+13$$ The associated polynomial will cut the $x$ axis in three locations. Figure 1 shows the situation for $y=x \times (x+1) \times (2x+3)$ where $x$=-1.5, -1 and 0.

Figure 1: Geogebra link

Let's try another example where a, b, c, d, e, f = 1, -1, 1, 1, 1 , 2. The initial terms generated will be of the form $(n-1) \times (n+1) \times (n+2)$ and are as follows (starting with $n=3$ because we want to avoid getting a 1 as a factor):

40, 90, 168, 280, 432, 630, 880, 1188, 1560, 2002, 2520, 3120, 3808, 4590, 5472, 6460, 7560, 8778, 10120, 11592, 13200, 14950, 16848, 18900, 21112, 23490, 26040, 28768, 31680, 34782

This sequence does not appear in the OEIS but it is recognised as being generated from a polynomial (I included the terms for $n=1$ and $n=2$ when searching in the OEIS). See Figure 2.

Figure 2

For a given number to be a pseudo-sphenic number, it must have more than three, not necessarily distinct, prime factors. For example, my earlier example of 27048: $$27048=2^3 \times 3 \times 7^2 \times 23=23 \times 24 \times 49$$ Sometimes the members of a polynomial sequence will be genuinely sphenic as with the case of: $$n \times (n+6) \times (n+12)$$ Figure 3 shows the presence of triplets of so-called "sexy" primes: (5, 11, 17), (11, 17, 23), (17, 23, 29) and (31, 37, 43).

Figure 3: permalink

Any sphenic number will be a member of some polynomial sequence. Let's arbitrarily choose the sphenic number formed by the prime factors 79, 101 and 197. We have: $$1571863=79 \times 101 \times 197$$ Let's subtract 77 from each number so that they become smaller. This gives us 2, 24 and 120 so that we consider the polynomial: $$(n+2) \times (n+24) \times (n+ 120)$$ Figure 4 shows the result when we generate the sequence up to 1571863.

Figure 4: permalink

Looking at the results we can see that:

• 106723 = 19 * 41 * 137
• 244807 = 31 * 53 * 149
• 906277 = 61 * 83 * 179
• 1571863 = 79 * 101 * 197

Thus the seemingly unrelated sphenic numbers 106723, 244807, 906277 and 1571863 are in fact very closely related because they all arise when different values of $n$ are assigned to the polynomial $(n+2) \times (n+24) \times (n+120)$, specifically $n=17, 29, 59$ and $77$. Thus a hidden link between sphenic numbers is revealed.