cp's OEIS Frontend

A239710 Primes of the form m = b^i + b^j + 1, where i > j > 0, b > 1.

%I A239710 #19 Oct 26 2014 04:59:59
%S A239710 7,11,13,19,31,37,41,43,67,73,97,109,131,137,151,157,193,211,223,241,
%T A239710 271,307,421,463,521,577,601,631,641,733,739,751,757,769,811,1033,
%U A239710 1123,1153,1303,1453,1483,1723,1741,1873,2053,2081,2113,2269,2551,2917,2971,3251,3307,3391,3541,3907,4099
%N A239710 Primes of the form m = b^i + b^j + 1, where i > j > 0, b > 1.
%C A239710 If m is a term, then there is a base b > 1 such that the base-b representation of m has digital sum = 3.
%C A239710 The base b for which m = b^i + b^j + 1 is not uniquely determined. Example: 13 = 2^3+2^2+1 = 3^2 +3^1+1.
%C A239710 Numbers m that satisfy m = b^i + b^j + 1 and b == 1 (mod 3) are not terms.
%H A239710 Hieronymus Fischer, <a href="/A239710/b239710.txt">Table of n, a(n) for n = 1..10000</a>
%e A239710 a(1) = 7, since 7 = 2^2 + 2^1 + 1 is prime.
%e A239710 a(2) = 11, since 11 = 2^3 + 2^1 + 1 is prime.
%e A239710 a(5) = 31, since 31 = 3^3 + 3^1 + 1 is prime.
%e A239710 a(10) = 73.
%e A239710 a(100) = 24181.
%e A239710 a(10^3) = 23160157.
%e A239710 a(10^4) = 7039461703.
%e A239710 a(10^5) = 1226630453623.
%e A239710 a(10^6) = 182489744292253.
%o A239710 (Smalltalk)
%o A239710 A239710
%o A239710   "Answers the n-th term of A239710.
%o A239710 Usage: n A239710
%o A239710 Answer: a(n)"
%o A239710   ^(self primesWhichAreDistinctPowersWithOffset: 1) at: self
%o A239710 -----------
%o A239710 (Smalltalk)
%o A239710 primesWhichAreDistinctPowersWithOffset: d
%o A239710   "Answers an array which hold the first n primes of the form b^i + b^j + d, i>j>0, where n is the receiver. Iterative calculation, b > 1.
%o A239710   Usage: n primesWhichAreDistinctPowersWithOffset: d
%o A239710 Answer: all terms < n"
%o A239710   | n terms m |
%o A239710   terms := OrderedCollection new.
%o A239710   n := self.
%o A239710   m := n squared * (n integerCeilLog: 2) * 2.
%o A239710   terms := m primesLTnWhichAreDistinctPowersWithOffset: d.
%o A239710   [terms size < n] whileTrue:
%o A239710     [m := 2 * m.
%o A239710     terms := m primesLTnWhichAreDistinctPowersWithOffset: d].
%o A239710   ^(terms copyFrom: 1 to: n) asArray
%o A239710 -----------
%o A239710 (Smalltalk)
%o A239710 primesLTnWhichAreDistinctPowersWithOffset: d
%o A239710   "Answers an array which hold the primes < n of the form b^i + b^j + d, i>j>0, where n is the receiver, b > 1.
%o A239710   Uses floorDistinctPowersWithOffset: d from A242100"
%o A239710   ^(self floorDistinctPowersWithOffset: d) select: [:i | i isPrime]
%Y A239710 Cf. A239708, A239709, A239711.
%Y A239710 Cf. A242100, A239712 - A239720.
%K A239710 nonn
%O A239710 1,1
%A A239710 _Hieronymus Fischer_, Mar 27 2014 and May 04 2014