A321058 - OEIS

%I #28 Sep 15 2022 11:54:28

%S 1,2,44,561329,9549959,42932384,13044904289,277344139214,

%T 2045466215756534,47702521115271164

%N Indices of records in A067849.

%C The corresponding record values are 2,4,6,7,8,9,10,12,13,14.

%C From _David A. Corneth_, Nov 10 2018: (Start)

%C Terms a(n) are of the form 3*k+2 for n > 1.

%C If 2^k - 1 is composite then a(n) is not divisible by any prime factor of 2^k-1 for n > k. So for example, gcd(a(n), 105) = 1 for n > 5. (End)

%C From _Glen Whitney_, Sep 14 2022: (Start)

%C Similarly to Corneth's observations, modulo any prime p, any residue for a(n) of the form 2^k - 1 mod p is forbidden for n greater than or equal to the number of such residues; for example a(n) may not be congruent to 0, 1, or 3 mod 7 for n >= 3.

%C For n > 2, if a(n) appears in this sequence, 2a(n) + 1 must appear in A057331. (End)

%e 2 is a term because A067849(2) = 4 > A067849(1) = 2.

%e 44 is a term because A067849(44) = 6 > A067869(2) = 4.

%o (PARI) b(n) = {my(nb = 0, newn); while (isprime(newn=2*n+1), nb++; n = newn); nb;} \\ A067849

%o lista(nn) = {my(mmax = -1, mm); for (n=1, nn, if ((mm=b(n)) > mmax, mmax = mm; print1(n, ", ")););} \\ _Michel Marcus_, Nov 10 2018

%Y Cf. A067849, A005602, A057331, A063378.

%K nonn,more

%O 1,2

%A _Torlach Rush_, Oct 26 2018

%E a(7) from _Amiram Eldar_, Nov 10 2018

%E a(8)-a(10) from A057331 by _Glen Whitney_, Sep 14 2022