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.

3, 11, 61, 521, 547, 683, 2731, 9091, 13421, 19141, 43691, 61681, 152381, 174763, 185641, 224071, 398581, 909091, 1151041, 1623931, 1824841, 2031671, 2796203, 3341101, 4778021, 5200081, 7027567, 8987221, 10678711, 15790321, 22796593, 25058741, 31224301, 32222107

Offset: 1

Author

Bernard Schott, Apr 29 2021

Keywords

• nonn
• base

Comments

The exponents k, q are necessarily primes.

Equivalently: primes of the form (c^k+1)/(c+1) that are not Brazilian: intersection of A059055 and A220627.

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.

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.

Examples

			3 = (2^3+1)/(2+1) is not Brazilian, hence 3 is a term.
11 = (2^5+1)/(2+1) is not Brazilian, hence 11 is a term.
547 = (3^7+1)/(3+1) is not Brazilian, hence 547 is a term.
9091 = (10^5+1)/(10+1) is not Brazilian, hence 9091 is a term.
		

Links

• H. Dubner and T. Granlund, Primes of the form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.

Crossrefs

Primes of the form (b^k-1)/(b-1) = A085104 (Brazilian primes).

Primes of the form (c^q+1)/(c+1) = A059055.

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).

Primes of the form (b^k-1)/(b-1) but not (c^q+1)/(c+1) = A225148.

Primes of the form (c^q+1)/(c+1) but not (b^k-1)/(b-1) = this sequence.

Primes neither of the form (c^q+1)/(c+1) nor (b^k-1)/(b-1) = A343775.

Programs

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); ); );
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);
isok(p) = isprime(p) && isc(p) && isnotb(p); \\ Michel Marcus, May 01 2021

Extensions

More terms from Michel Marcus, Apr 30 2021