cp's OEIS Frontend

A239713 Primes of the form m = 3^i + 3^j - 1, where i > j >= 0.

%I A239713 #25 Mar 17 2025 07:02:12
%S A239713 3,11,29,83,89,107,251,269,809,971,2213,2267,6563,6569,6803,8747,
%T A239713 19709,19763,20411,59051,65609,177173,183707,531521,538001,590489,
%U A239713 1594331,1594403,1595051,1596509,4782971,4782977,4783697,14348909,14349149,14526053,14880347
%N A239713 Primes of the form m = 3^i + 3^j - 1, where i > j >= 0.
%C A239713 The base-3 representation of a term 3^i + 3^j - 1 has base-3 digital sum = 1 + 2*j == 1 (mod 2).
%C A239713 In base-3 representation the first terms are 10, 102, 1002, 10002, 10022, 10222, 100022, 100222, 1002222, 1022222, 10000222, 10002222, 100000002, 100000022, 100022222, 102222222, 1000000222, 1000002222, 1000222222, 10000000002, 10022222222, 100000000222, 100022222222, ...
%H A239713 Hieronymus Fischer, <a href="/A239713/b239713.txt">Table of n, a(n) for n = 1..131</a> [a(123) corrected by _Georg Fischer_, Dec 22 2024]
%e A239713 a(1) = 3, since 3 = 3^1 + 3^0 - 1 is prime.
%e A239713 a(5) = 89, since 89 = 3^4 + 3^2 - 1 is prime.
%p A239713 select(isprime, [seq(seq(3^i+3^j-1, j=0..i-1), i=1..25)])[];  # _Alois P. Heinz_, Dec 22 2024
%t A239713 Select[Flatten[Table[3^i + 3^j - 1, {i, 1, 25}, {j, 0, i - 1}]], PrimeQ] (* _Jean-François Alcover_, Mar 17 2025, after _Alois P. Heinz_ *)
%o A239713 (Smalltalk)
%o A239713 A239713
%o A239713 "Answers the n-th term of A239713.
%o A239713   Usage: n A239713
%o A239713   Answer: a(n)"
%o A239713   | a b i j k p q terms |
%o A239713   terms := OrderedCollection new.
%o A239713   k := 0.
%o A239713   b := 3.
%o A239713   p := b.
%o A239713   i := 1.
%o A239713   [k < self] whileTrue:
%o A239713          [j := 0.
%o A239713          q := 1.
%o A239713          [j < i and: [k < self]] whileTrue:
%o A239713                    [a := p + q - 1.
%o A239713                    a isPrime
%o A239713                         ifTrue:
%o A239713                             [k := k + 1.
%o A239713                             terms add: a].
%o A239713                    q := b * q.
%o A239713                    j := j + 1].
%o A239713          i := i + 1.
%o A239713          p := b * p].
%o A239713      ^terms at: self
%o A239713 [by _Hieronymus Fischer_, Apr 14 2014]
%o A239713 --------------------
%o A239713 (Smalltalk)
%o A239713 A239713
%o A239713 "Version 2: Answers the n-th term of A239713.
%o A239713   Uses distinctPowersOf: b from A018900
%o A239713   Usage: n A239713
%o A239713   Answer: a(n)”
%o A239713   | a k n terms |
%o A239713   terms := OrderedCollection new.
%o A239713   n := 1.
%o A239713   k := 0.
%o A239713   [k < self] whileTrue:
%o A239713          [(a:= (n distinctPowersOf: 3) - 1)
%o A239713               isPrime ifTrue:    [k := k + 1.
%o A239713                                  terms add: a].
%o A239713               n := n + 1].
%o A239713   ^terms at: self
%o A239713 [by _Hieronymus Fischer_, Apr 22 2014]
%o A239713 -----------
%o A239713 (Smalltalk)
%o A239713 A239713
%o A239713   "Version 3: Answer an array of the first n terms of A239713.
%o A239713   Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
%o A239713   Usage: n A239713
%o A239713   Answer: #(3 11 29 ... ) [a(1) ... a(n)]”
%o A239713   ^self primesWhichAreDistinctPowersOf: 3 withOffset: -1
%o A239713 [by _Hieronymus Fischer_, Apr 22 2014]
%Y A239713 Cf. A018900, A239709, A239712 (base 2), A239714 (base 4), A239715 (base 5), A239716 (base 6), A239717 (base 7), A239718 (base 8), A239719 (base 9), A239720 (base 10).
%K A239713 nonn
%O A239713 1,1
%A A239713 _Hieronymus Fischer_, Mar 28 2014