A187823 Primes of the form (p^x - 1)/(p^y - 1), where p is prime, y > 1, and y is the largest proper divisor of x.

5, 17, 73, 257, 757, 65537, 262657, 1772893, 4432676798593, 48551233240513, 378890487846991, 3156404483062657, 17390284913300671, 280343912759041771, 319913861581383373, 487014306953858713, 5559917315850179173, 7824668707707203971, 8443914727229480773, 32564717507686012813

Offset

1,1

Comments

Complement of A023195 relative to A003424.

Only eight primes of this form don't exceed 1.275*10^10 (see Bateman and Stemmler):

(1) three of the form (p^9 - 1)/(p^3 - 1): 73 (p=2), 757 (p=3), 1772893 (p=11);

(2) four of the form (2^x - 1)/(2^y - 1) with x = 2y: 5 (x=4), 17 (x=8), 257 (x=16), 65537 (x=32); and

(3) the prime 262657 = (2^27 - 1)/(2^9 - 1).

Some of these prime numbers are not Brazilian, these are Fermat primes > 3: 5, 17, 257, 65537, so they are in A220627.

The other primes are Brazilian so they are in A085104, example: (p^9 - 1)/(p^3 - 1) = 111_{p^3} with 73 = 111_8, 757 = 111_27, 1772893 = 111_1331, also 262657 = 111_512 [See section V.4 of Quadrature article in Links] (comment improved in Mar 03 2023).

Comments from Don Reble, Jul 28 2022 (Start)

This is an easy sequence that looks hard.

Note that x must be a power of a prime; otherwise (p^x-1)/(p^y-1) has too many cyclotomic factors.

Almost all values are (p^9-1)/(p^3-1). The exceptions below 10^45

are the Fermat primes 5, 17, 257, 65537 and also

262657, 4432676798593, 5559917315850179173,

227376585863531112677002031251,

467056170954468301850494793701001,

36241275390490156321975496980895092369525753,

284661951906193731091845096405947222295673201 (see examples).

(End)

Links

Don Reble, Table of n, a(n) for n = 1..50000

Paul T. Bateman and Rosemarie M. Stemmler, Waring's problem for algebraic number fields and primes of the form (p^r-1)/(p^d-1), Illinois J. Math. 6 (1962), pp. 142-156.

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Index entries for sequences related to Brazilian numbers.

Example

5 = (2^4 - 1)/(2^2 - 1)= 11_{2^2} = 11_4.
17 = (2^8 - 1)/(2^4 - 1) = 11_{2^4} = 11_16.
257 = (2^16 - 1)/(2^8 - 1) = 11_{2^8} = 11_256.
757 = (3^9 - 1)/(3^3 - 1) = 111_{3^3} = 111_27.
262657 = (2^27 - 1)/(2^9 - 1) = 111_{2^9} = 111_512.
655357 = (2^32 - 1)/(2^16 - 1) = 11_{2^16} = 11_655356.
4432676798593 = (2^49 - 1)/(2^7 - 1) = 1111111_{2^7} = 1111111_128.
5559917315850179173 = (11^27 - 1)/(11^9 - 1) = 111_{11^3} = 111_1331.
227376585863531112677002031251 = (5^49 - 1)/(5^7 - 1) = 1111111_{5^7}.
467056170954468301850494793701001 = (43^25 - 1)/(43^5 - 1) = 11111_{43^5}.
36241275390490156321975496980895092369525753 = (263^27 - 1)/(263^9 - 1).
284661951906193731091845096405947222295673201 = (167^25 - 1)/(167^5 - 1).

Crossrefs

Equals A003424 \ A023195.

Cf. A085104, A220627.

Sequence in context: A146577 A211405 A354040 * A297625 A084777 A149717

Adjacent sequences: A187820 A187821 A187822 * A187824 A187825 A187826

Keyword

nonn

Author

Bernard Schott, Dec 27 2012

Extensions

a(9)-a(16) from Don Reble, Jul 28 2022

a(17)-a(20) from Don Reble, Mar 21 2023

Status

approved