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.