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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.

%I A187823 #68 Mar 29 2023 08:59:26
%S A187823 5,17,73,257,757,65537,262657,1772893,4432676798593,48551233240513,
%T A187823 378890487846991,3156404483062657,17390284913300671,
%U A187823 280343912759041771,319913861581383373,487014306953858713,5559917315850179173,7824668707707203971,8443914727229480773,32564717507686012813
%N 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.
%C A187823 Complement of A023195 relative to A003424.
%C A187823 Only eight primes of this form don't exceed 1.275*10^10 (see Bateman and Stemmler):
%C A187823 (1) three of the form (p^9 - 1)/(p^3 - 1): 73 (p=2), 757 (p=3), 1772893 (p=11);
%C A187823 (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
%C A187823 (3) the prime 262657 = (2^27 - 1)/(2^9 - 1).
%C A187823 Some of these prime numbers are not Brazilian, these are Fermat primes > 3: 5, 17, 257, 65537, so they are in A220627.
%C A187823 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).
%C A187823 Comments from _Don Reble_, Jul 28 2022 (Start)
%C A187823 This is an easy sequence that looks hard.
%C A187823 Note that x must be a power of a prime; otherwise (p^x-1)/(p^y-1) has too many cyclotomic factors.
%C A187823 Almost all values are (p^9-1)/(p^3-1). The exceptions below 10^45
%C A187823     are the Fermat primes 5, 17, 257, 65537 and also
%C A187823     262657, 4432676798593, 5559917315850179173,
%C A187823     227376585863531112677002031251,
%C A187823     467056170954468301850494793701001,
%C A187823     36241275390490156321975496980895092369525753,
%C A187823     284661951906193731091845096405947222295673201 (see examples).
%C A187823 (End)
%H A187823 Don Reble, <a href="/A187823/b187823.txt">Table of n, a(n) for n = 1..50000</a>
%H A187823 Paul T. Bateman and Rosemarie M. Stemmler, <a href="https://doi.org/10.1215/ijm/1255631815">Waring's problem for algebraic number fields and primes of the form (p^r-1)/(p^d-1)</a>, Illinois J. Math. 6 (1962), pp. 142-156.
%H A187823 Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
%H A187823 <a href="/index/Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>.
%e A187823 5 = (2^4 - 1)/(2^2 - 1)= 11_{2^2} = 11_4.
%e A187823 17 = (2^8 - 1)/(2^4 - 1) = 11_{2^4} = 11_16.
%e A187823 257 = (2^16 - 1)/(2^8 - 1) = 11_{2^8} = 11_256.
%e A187823 757 = (3^9 - 1)/(3^3 - 1) = 111_{3^3} = 111_27.
%e A187823 262657 = (2^27 - 1)/(2^9 - 1) = 111_{2^9} = 111_512.
%e A187823 655357 = (2^32 - 1)/(2^16 - 1) = 11_{2^16} = 11_655356.
%e A187823 4432676798593 = (2^49 - 1)/(2^7 - 1) = 1111111_{2^7} = 1111111_128.
%e A187823 5559917315850179173 = (11^27 - 1)/(11^9 - 1) = 111_{11^3} = 111_1331.
%e A187823 227376585863531112677002031251 = (5^49 - 1)/(5^7 - 1) = 1111111_{5^7}.
%e A187823 467056170954468301850494793701001 = (43^25 - 1)/(43^5 - 1) = 11111_{43^5}.
%e A187823 36241275390490156321975496980895092369525753 = (263^27 - 1)/(263^9 - 1).
%e A187823 284661951906193731091845096405947222295673201 = (167^25 - 1)/(167^5 - 1).
%Y A187823 Equals A003424 \ A023195.
%Y A187823 Cf. A085104, A220627.
%K A187823 nonn
%O A187823 1,1
%A A187823 _Bernard Schott_, Dec 27 2012
%E A187823 a(9)-a(16) from _Don Reble_, Jul 28 2022
%E A187823 a(17)-a(20) from _Don Reble_, Mar 21 2023