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A342067

Primes p such that p^11 - 1 has 8 divisors.

3, 467, 2039, 4679, 5399, 5939, 6899, 8783, 12347, 16487, 18443, 23879, 25583, 33647, 35879, 36299, 44819, 47207, 53147, 57119, 67499, 74507, 90239, 93287, 96059, 119759, 125003, 133499, 135119, 136223, 157019, 159539, 164999, 165059, 168887, 178799, 188159

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OFFSET

1,1

COMMENTS

Conjecture: sequence is infinite.

The only primes p such that p^11 - 1 has fewer than A309906(11)=8 divisors are 2 and 5.

p^11 - 1 = (p-1)*(p^10 + p^9 + p^8 + p^7 + p^6 + p^5 + p^4 + p^3 + p^2 + p + 1).

For every term p, p^11 - 1 is of the form 2*q*r, where q and r are distinct odd primes. With the exception of p=a(1)=3, each term p is a number such that (p-1)/2 and (p^10 + p^9 + p^8 + ... + p^2 + p + 1) are primes.

LINKS

Table of n, a(n) for n=1..37.

EXAMPLE

p =

n a(n) factorization of p^11 - 1

- ---- ------------------------------------------------

1 3 2 * 23 * 3851

2 467 2 * 233 * 494424256962371823779424877

3 2039 2 * 1019 * 1242754384106847037173120489949801

4 4679 2 * 2339 * 5030640462820574591105701447273296601

PROG

(PARI) isok(p) = isprime(p) && (numdiv(p^11-1) == 8); \\ Michel Marcus, Feb 28 2021