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

Author

Bernard Schott, Dec 27 2012

Keywords

• nonn

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)

Examples

			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).
		

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.

Crossrefs

Equals A003424 \ A023195.

Cf. A085104, A220627.

Extensions

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

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