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Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer to 2^n/3.
(Formerly M2482 N0983)
+10
710
0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531, 5726623061, 11453246123
COMMENTS
All prime Jacobsthal numbers A049883[n] = {3, 5, 11, 43, 683, 2731, 43691, ...} have prime indices except for a(4) = 5. All prime Jacobsthal numbers with prime indices (all but a(4) = 5) are of the form (2^p + 1)/3 - the Wagstaff primes A000979[n]. Indices of prime Jacobsthal numbers are listed in A107036[n] = {3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, ...}. For n>1 A107036[n] = A000978[n] Numbers n such that (2^n + 1)/3 is prime. - Alexander Adamchuk, Oct 03 2006
Wagstaff primes: primes of form (2^p + 1)/3.
(Formerly M2896 N1161)
+10
29
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, 62357403192785191176690552862561408838653121833643
Smallest prime of the form (q^p+1)/(q+1), where p = prime(n) and q is also prime (q = A123627(n)); or 1 if such a prime does not exist.
+10
5
1, 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 402488219476647465854701, 715827883, 10300379826060720504760427912621791994517454717, 254760179343040585394724919772965278539769280548173566545431025735121201
List of primitive prime divisors of the Jacobsthal numbers A001045 in their order of occurrence.
+10
4
3, 5, 11, 7, 43, 17, 19, 31, 683, 13, 2731, 127, 331, 257, 43691, 73, 174763, 41, 5419, 23, 89, 2796203, 241, 251, 4051, 8191, 87211, 29, 113, 59, 3033169, 151, 715827883, 65537, 67, 20857, 131071, 281, 86171, 37, 109, 1777, 25781083, 524287, 22366891, 61681, 83
Numbers n such that (2^p + 1)/3 is prime, where p is the n-th prime.
+10
2
2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, 31, 39, 43, 46, 65, 69, 126, 267, 380, 495, 762, 1285, 1304, 1364, 1479, 1697, 4469, 8135, 9193, 11065, 11902, 12923, 13103, 23396, 23642, 31850, 77509, 285228
COMMENTS
Also prime(a(n)) are the indices of prime Jacobsthal numbers ( A001045) with prime indices. Primes in the Jacobsthal sequence are listed in A049883.
Jacobsthal numbers which are semiprimes.
+10
2
21, 85, 341, 5461, 22369621, 178956971, 5726623061, 45812984491, 91625968981, 733007751851, 46912496118443, 187649984473771, 3002399751580331, 1537228672809129301, 49191317529892137643, 787061080478274202283, 3148244321913096809131
Primes q such that (2^p + 1)/3 is prime, where p = Prime[q]; or primes in A123176[n].
+10
1
2, 3, 5, 7, 11, 31, 43, 1697, 12923, 13103, 77509
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