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A309906 a(n) is the smallest number of divisors of p^n - 1 that can occur for arbitrarily large primes p. 0
4, 32, 8, 160, 8, 384, 8, 384, 16, 256, 8, 7680, 8, 128, 32, 1792, 8, 4096, 8, 3840, 32, 256, 8, 36864, 16, 128, 32, 2560, 8, 24576, 8, 4096, 32, 128, 32, 327680, 8, 128, 32, 36864, 8, 18432, 8, 2560, 128, 256, 8, 344064, 16, 1024, 32, 2560, 8, 20480, 32 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For each prime q, every number k that has exactly q divisors is a prime power k = p^(q-1) for some prime p. As a result, a(q-1) can be useful in identifying numbers of the form p^(q-1) - 1 that are terms of A161460 (see Example section).

From Bernard Schott, Aug 22 2019: (Start)

For n prime >= 3, a(n) = 8;

for n = q^2, q prime >= 3, a(n) = 16. (End)

LINKS

Table of n, a(n) for n=1..55.

FORMULA

a(n) = A000005(A079612(n))*2^A000005(n).

EXAMPLE

a(1) = 4: The only primes p for which p-1 has fewer than 4 divisors are 2, 3, and 5; for all primes p > 5, p-1 has at least 4 divisors, and the terms in A005385 (Safe primes) except 5 are primes p such that p-1 has exactly 4 divisors.

a(2) = 32: p^2 - 1 = (p-1)*(p+1) has fewer than 32 divisors only for p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 47, and 73; for all primes p such that the product of the 3-smooth parts of p-1 and p+1 is 24 and p-1 and p+1 each have one prime factor > 3, p^2 - 1 has exactly 32 divisors.

a(4) = 160: primes p such that p^4 - 1 has exactly 160 divisors are plentiful, but only p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 59, 61, 71, 79, and 101 yield tau(p^4 - 1) < 160. Of these, p = 13, 29, 59, and 61 all give tau(p^4 - 1) = 80; 37 and 101 both give 120 divisors; and 41 and 71 both give 144. For each of the ten remaining primes (p = 2, 3, 5, 7, 11, 17, 19, 23, 31, 79), the value of tau(p^4 - 1) is unique, so each of those ten values of p^4 - 1 is a term in A161460.

PROG

(PARI) f(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)); ); ); res; ); } \\ A079612

a(n) = numdiv(f(n))*2^numdiv(n); \\ Michel Marcus, Aug 22 2019

CROSSREFS

Cf. A000005, A005385, A006863, A079612, A161460.

Sequence in context: A084764 A061789 A103909 * A196247 A196250 A290809

Adjacent sequences:  A309903 A309904 A309905 * A309907 A309908 A309909

KEYWORD

nonn

AUTHOR

Jon E. Schoenfield, Aug 21 2019

STATUS

approved

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Last modified October 22 18:08 EDT 2019. Contains 328319 sequences. (Running on oeis4.)