cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A239709 Primes of the form m = b^i + b^j - 1, where i > j > 0, b > 1.

Original entry on oeis.org

5, 11, 17, 19, 23, 29, 41, 47, 67, 71, 79, 83, 89, 107, 109, 131, 149, 181, 191, 239, 251, 257, 263, 269, 271, 349, 379, 383, 419, 461, 599, 701, 809, 811, 929, 971, 991, 1009, 1031, 1039, 1087, 1151, 1259, 1279, 1301, 1451, 1481, 1511, 1559, 1721, 1871, 1979, 2063, 2069, 2111, 2161, 2213, 2267, 2351, 2549, 2861, 2939, 2969, 3079, 3191, 3389
Offset: 1

Views

Author

Hieronymus Fischer, Mar 27 2014

Keywords

Comments

If m is a term, then there is a base b > 1 such that the base-b representation of m has digital sum = 1 + j*(b-1) == 1 (mod (b-1)).
The base b for which m = b^i + b^j - 1 is not uniquely determined. Example: 11 = 2^3+2^2-1 = 3^2 +3^1-1.
Numbers m which satisfy m = b^i + b^j - 1 with odd i and j and b == 2 (mod 3) are not terms. Example: 12189 = 23^3 + 23^1 - 1 is not a prime.

Examples

			a(1) = 5, since 5 = 2^2 + 2^1 - 1 is prime.
a(2) = 11, since 11 = 2^3 + 2^2 - 1 is prime.
a(6) = 29, since 29 = 3^3 + 3^1 - 1 is prime.
a(10^1) = 71.
a(10^2) = 13109.
a(10^3) = 9336079.
a(10^4) = 2569932329.
a(10^5) = 455578426189.
a(10^6) = 68543190483641.

Links

Crossrefs

Cf. A239710, A239711, A239712 - A239720.
Cf. A239708, A018900.

Programs

  • Smalltalk
    A239709
"Answers the n-th term of A239709.
  Iterative calculation using A239709_termsLTn.
  Usage: n A239709
  Answer: a(n)"
  | n terms m |
  terms := SortedCollection new.
  n := self.
  m := (n prime // 2) squared.
  terms := m A239709_termsLTn.
  [terms size < n] whileTrue:
         [m := 2 * m.
         terms := m A239709_termsLTn].
  ^terms at: n
  "Remark: A last line of
  ^terms copyFrom: 1 to: n
  answers an array of the first n terms"
[by_Hieronymus Fischer_, Apr 14 2014]
-----------
  • Smalltalk
    A239709_termsLTn
  "Answers all the terms of A239709 which are < n.
  Direct processing by scanning the scanning the bases b in increasing order, up to b = sqrt(n), and calculating the numbers b^i + b^j - 1.
  Usage: n A239709_termsLTn
  Answer: #(5 11 17 19 23 ...) [terms < n]"
  | bmax p q n m terms a |
  terms := OrderedCollection new.
  n := self.
  bmax := n sqrtTruncated.
  2 to: bmax
     do:
         [:b |
         m := 1 + (n floorLog: b).
         p := b.
         2 to: m
              by: 1
              do:
                   [:i |
                   p := b * p.
                   q := b.
                   1 to: i - 1
                        by: 1
                        do:
                            [:j |
                            a := p + q - 1.
                            a < n ifTrue: [a isPrime ifTrue: [terms add: a]].
                            q := b * q]]].
  ^terms asSet asArray sorted
[by_Hieronymus Fischer_, Apr 14 2014]
-----------
  • Smalltalk
    A239709nTerms
  "Alternative version: Answers the first n terms of A239709. Direct calculation by scanning the numbers b^i + b^j - 1 in increasing order.
  Usage: n A239709
  Answer: a(n)"
  | a amax an b bmax k terms p q p_i q_j a_b amin bamin |
  terms := SortedCollection new.
  p_i := OrderedCollection new.
  q_j := OrderedCollection new.
  a_b := OrderedCollection new.
  p_i add: 1.
  q_j add: 1.
  a_b add: 1.
  k := 0.
  b := 2.
  bmax := b.
  p := b * b.
  q := b.
  a := p + q - 1.
  p_i add: p.
  q_j add: q.
  a_b add: a.
  amax := 2 * (b + 1) + a.
  an := 0.
  [(k < self and: [a < amax]) or: [a < an]] whileTrue:
         [[(k < self and: [a < amax]) or: [a < an]] whileTrue:
                   [[q < p and: [(k < self and: [a < amax]) or: [a < an]]] whileTrue:
                            [a isPrime2
                                 ifTrue:
                                      [(terms includes: a)
                                          ifFalse:
                                               [k := k + 1.
                                               terms add: a.
                                               k >= self ifTrue: [an := terms at: self]]].
                            q := b * q.
                            a := p + q - 1].
                   p = q
                        ifTrue:
                            [p := b * p.
                            q := b.
                            a := p + q - 1].
                   p_i at: b put: p.
                   q_j at: b put: q.
                   a_b at: b put: a].
         amin := a.
         2 to: b - 1
              do:
                   [:bb |
                   (a_b at: bb) < amin
                        ifTrue:
                            [amin := a_b at: bb.
                            bamin := bb]].
         b + 1 to: bmax
              do:
                   [:bb |
                   (a_b at: bb) < amin
                        ifTrue:
                            [amin := a_b at: bb.
                            bamin := bb]].
         amin < (a min: amax)
              ifTrue:
                   [b := bamin.
                   p := p_i at: b.
                   q := q_j at: b.
                   a := a_b at: b]
              ifFalse:
                   [bmax := bmax + 1.
                   b := bmax.
                   p := b * b.
                   q := b.
                   a := p + q - 1.
                   p_i add: p.
                   q_j add: q.
                   a_b add: a.
                   amax := 2 * (b + 1) + a max: amax]].
  ^terms copyFrom: 1 to: self
[by_Hieronymus Fischer_, Apr 20 2014]

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

Original entry on oeis.org

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

Views

Author

Hieronymus Fischer, Mar 27 2014 and May 04 2014

Keywords

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

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]
Showing 1-2 of 2 results.

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