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A343774 Primes of the form (c^k+1)/(c+1) not having a representation in the form (b^q-1)/(b-1), where b, c > 1 and k, q > 2.

%I A343774 #28 May 01 2021 13:38:44
%S A343774 3,11,61,521,547,683,2731,9091,13421,19141,43691,61681,152381,174763,
%T A343774 185641,224071,398581,909091,1151041,1623931,1824841,2031671,2796203,
%U A343774 3341101,4778021,5200081,7027567,8987221,10678711,15790321,22796593,25058741,31224301,32222107
%N A343774 Primes of the form (c^k+1)/(c+1) not having a representation in the form (b^q-1)/(b-1), where b, c > 1 and k, q > 2.
%C A343774 The exponents k, q are necessarily primes.
%C A343774 Equivalently: primes of the form (c^k+1)/(c+1) that are not Brazilian: intersection of A059055 and A220627.
%C A343774 Except for 3 where k = 3, all the terms of this sequence are of the form (c^k+1)/(c+1) with k prime >= 5.
%C A343774 The only known prime of this form with k prime >= 5 that is not present is 43 = (2^7+1)/(2+1) because also 43 = (7^3+1)/(7+1) = (6^3-1)/(6-1) = 111_6, so 43 belongs to A002383.
%H A343774 H. Dubner and T. Granlund, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.html">Primes of the form (b^n+1)/(b+1)</a>, J. Integer Sequences, 3 (2000), #P00.2.7.
%e A343774 3 = (2^3+1)/(2+1) is not Brazilian, hence 3 is a term.
%e A343774 11 = (2^5+1)/(2+1) is not Brazilian, hence 11 is a term.
%e A343774 547 = (3^7+1)/(3+1) is not Brazilian, hence 547 is a term.
%e A343774 9091 = (10^5+1)/(10+1) is not Brazilian, hence 9091 is a term.
%o A343774 (PARI) isc(p) = for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1); ); );
%o A343774 isnotb(p) = for (b=2, p-1, my(d=digits(p, b), md=vecmin(d)); if ((#d > 2) && (md == 1) && (vecmax(d) == 1), return (0)); ); return (1);
%o A343774 isok(p) = isprime(p) && isc(p) && isnotb(p); \\ _Michel Marcus_, May 01 2021
%Y A343774 Primes of the form (b^k-1)/(b-1) = A085104 (Brazilian primes).
%Y A343774 Primes of the form (c^q+1)/(c+1) = A059055.
%Y A343774 Primes of the form (b^k-1)/(b-1) and (c^q+1)/(c+1): A002383 \ {3} is a subsequence, but, maybe the intersection (conjecture).
%Y A343774 Primes of the form (b^k-1)/(b-1) but not (c^q+1)/(c+1) = A225148.
%Y A343774 Primes of the form (c^q+1)/(c+1) but not (b^k-1)/(b-1) = this sequence.
%Y A343774 Primes neither of the form (c^q+1)/(c+1) nor (b^k-1)/(b-1) = A343775.
%K A343774 nonn,base
%O A343774 1,1
%A A343774 _Bernard Schott_, Apr 29 2021
%E A343774 More terms from _Michel Marcus_, Apr 30 2021