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

7, 11, 13, 19, 31, 37, 41, 43, 67, 73, 97, 109, 131, 137, 151, 157, 193, 211, 223, 241, 271, 307, 421, 463, 521, 577, 601, 631, 641, 733, 739, 751, 757, 769, 811, 1033, 1123, 1153, 1303, 1453, 1483, 1723, 1741, 1873, 2053, 2081, 2113, 2269, 2551, 2917, 2971, 3251, 3307, 3391, 3541, 3907, 4099

Offset: 1

Author

Hieronymus Fischer, Mar 27 2014 and May 04 2014

Keywords

• nonn

Comments

If m is a term, then there is a base b > 1 such that the base-b representation of m has digital sum = 3.

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.

Numbers m that satisfy m = b^i + b^j + 1 and b == 1 (mod 3) are not terms.

Examples

			a(1) = 7, since 7 = 2^2 + 2^1 + 1 is prime.
a(2) = 11, since 11 = 2^3 + 2^1 + 1 is prime.
a(5) = 31, since 31 = 3^3 + 3^1 + 1 is prime.
a(10) = 73.
a(100) = 24181.
a(10^3) = 23160157.
a(10^4) = 7039461703.
a(10^5) = 1226630453623.
a(10^6) = 182489744292253.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A239708, A239709, A239711.

Cf. A242100, A239712 - A239720.

Programs

Smalltalk

A239710
  "Answers the n-th term of A239710.
Usage: n A239710
Answer: a(n)"
  ^(self primesWhichAreDistinctPowersWithOffset: 1) at: self
-----------

Smalltalk

primesWhichAreDistinctPowersWithOffset: d
  "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.
  Usage: n primesWhichAreDistinctPowersWithOffset: d
Answer: all terms < n"
  | n terms m |
  terms := OrderedCollection new.
  n := self.
  m := n squared * (n integerCeilLog: 2) * 2.
  terms := m primesLTnWhichAreDistinctPowersWithOffset: d.
  [terms size < n] whileTrue:
    [m := 2 * m.
    terms := m primesLTnWhichAreDistinctPowersWithOffset: d].
  ^(terms copyFrom: 1 to: n) asArray
-----------

Smalltalk

primesLTnWhichAreDistinctPowersWithOffset: d
  "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.
  Uses floorDistinctPowersWithOffset: d from A242100"
  ^(self floorDistinctPowersWithOffset: d) select: [:i | i isPrime]