A350780 Numbers that are the number of divisors of p^2 - 1 for some prime p.

2, 4, 8, 10, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 70, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 182, 184, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228

Offset

1,1

Comments

For all primes p > 73, tau(p^2 - 1) >= A309906(2) = 32.

Links

Table of n, a(n) for n=1..58.

Example

184 is a term: p = 111149057 is a prime, and p^2 - 1 = (p-1)*(p+1) = 2^22 * 3 * 53 * 18524843, which has 23*2*2*2 = 184 divisors.
190 is not a term: 190 = 2 * 5 * 19, so a number with 190 divisors must be of the form q^189, q^94 * r, q^37 * r^4, q^18 * r^9, or q^18 * r^4 * s, and for every prime p > 3, p^2 - 1 is a multiple of 24 = 2^3 * 8, so all the forms with 190 divisors are easily ruled out except for q^18 * r^4 * s. If p^2 - 1 = q^18 * r^4 * s, then it's one of the products 2^18 * 3^4 * s, 2^18 * r^4 * 3, 3^18 * 2^4 * s, or q^18 * 2^4 * 3. Each of these can be shown to be impossible by examining all possible ways of factoring the product into two even factors (p-1 and p+1) that differ by exactly two.

Crossrefs

Cf. A000005, A309906, A341660.

Sequence in context: A336659 A341655 A171757 * A128106 A354776 A125021

Adjacent sequences: A350777 A350778 A350779 * A350781 A350782 A350783

Keyword

nonn

Author

Jon E. Schoenfield, May 02 2022

Status

approved