A239712 -id:A239712 - cp's OEIS frontend

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

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

Hieronymus Fischer, Mar 27 2014

nonn

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.

			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.

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

Cf. A239710, A239711, A239712 - A239720.

Cf. A239708, A018900.

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]
-----------

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]
-----------

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]

A239720 Primes of the form m = 10^i + 10^j - 1, where i > j >= 0.

Original entry on oeis.org

109, 1009, 10009, 10099, 100999, 1000099, 1000999, 1000000009, 1000009999, 1000099999, 1009999999, 10000000999, 10000099999, 10999999999, 100999999999, 1000000009999, 1000000999999, 1099999999999, 10000000000099, 10009999999999

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

Numbers with the first digit 1 followed by at least one 0-digit and ending with a number > 0 of trailing 9-digits.
The digital sum of a term 10^i + 10^j - 1 is = 1 + 9*j == 1 (mod 9).
Numbers m that satisfy m = 10^i + 10^j + 1 are never primes, since the digital sum of m is 3, and thus, m is divisible by 3.

			a(1) = 109, since 109 = 10^2 + 10^1 - 1 is prime.
a(2) = 1009, since 1009 = 10^3 + 10^1 - 1 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000 (First 44 terms from Hieronymus Fischer)

Select[Flatten[Table[10^i+10^j-1,{i,0,20},{j,0,i-1}]],PrimeQ] (* Harvey P. Dale, Jan 30 2017 *)

A239720
  "Answer the n-th term of A239720.
  Usage: n A239720
  Answer: a(n)"
  | a b i j k p q terms |
  terms := OrderedCollection new.
  k := 0.
  b := 10.
  p := b.
  i := 1.
  [k < self] whileTrue:
         [j := 0.
         q := 1.
         [j < i and: [k < self]] whileTrue:
                   [a := p + q - 1.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q.
                   j := j + 1].
         i := i + 1.
         p := b * p].
  ^terms at: self
--------------------

A239720
  "Version2: Answer an array of the first n terms of A239720.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239720
  Answer: #(109 1009 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 10 withOffset: -1

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

Hieronymus Fischer, Mar 27 2014 and May 04 2014

nonn

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.

			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.

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

Cf. A239708, A239709, A239711.

Cf. A242100, A239712 - A239720.

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

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
-----------

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]

A239719 Primes of the form m = 9^i + 9^j - 1, where i > j >= 0.

Original entry on oeis.org

89, 809, 6569, 65609, 531521, 538001, 590489, 4782977, 4783697, 47829689, 3486784409, 3491567369, 3529831121, 31768480097, 34867844009, 282430067921, 285916320881, 313810596089, 2541865834889, 22877179875449, 25418658283289

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

The base-9 representation of a term 9^i + 9^j - 1 has base-9 digital sum = 1 + 8*j == 1 (mod 8).

In base-9 representation the first terms are 108, 1088, 10008, 108888, 1000088, 1008888, 1088888, 10000008, 10000888, 108888888, 10000000008, 10008888888, 10088888888, 100888888888, ...

			a(1) = 89, since 89 = 9^2 + 9^1 - 1 is prime.
a(2) = 809, since 809 = 9^3 + 9^2 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..40

Select[Flatten[Table[9^i+9^j-1,{i,0,20},{j,0,i-1}]],PrimeQ] (* Harvey P. Dale, Jun 02 2023 *)

A239719
  "Answer an array of the first n terms of A239719.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239719
  Answer: #(89 809 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 9 withOffset: -1

A239711 Twin primes of the form m = b^i + b^j +- 1, where i > j > 0, b > 1.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 71, 73, 107, 109, 149, 151, 191, 193, 239, 241, 269, 271, 419, 421, 461, 463, 599, 601, 809, 811, 1031, 1033, 1151, 1153, 1301, 1303, 1451, 1453, 1481, 1483, 1721, 1723, 1871, 1873, 2111, 2113, 2267, 2269, 2549, 2551, 2969, 2971, 3389, 3391, 3539, 3541

Offset: 1

Hieronymus Fischer, Mar 27 2014 and May 04 2014

nonn

(a(2k-1), a(2k)), k > 0, form pairs of twin primes.
Numbers m that satisfy m = b^i + b^j + 1 and b == 1 (mod 3) and those that satisfy m = b^i + b^j - 1 with odd i and j and b == 2 (mod 3) are never terms, since they are divisible by 3. It follows that no numbers 4^i + 4^j +- 1, or 7^i + 7^j +- 1, or 10^i + 10^j +- 1, ... can be terms. Also, no numbers 5^(2m-1) + 5^(2k-1) +- 1, or 8^(2m-1) + 8^(2k-1) +- 1, or 11^(2m-1) + 11^(2k-1) +- 1, ... with m > k > 0, can be terms.
Example 1: 10^6 + 10^4 + 1 = 1010001 is not a term, since 10 == 1 (mod 3); certainly, 1010001 = 3*336667.
Example 2: 8^9 + 8^7 - 1 = 136314879 is not a term, since 8 == 2 (mod 3) and i, j odd; certainly 136314879 = 3*45438293.

			a(1) = 5, since 5 = 2^2 + 2^1 - 1 is prime.
a(2) = 7, since 7 = 2^3 + 2^1 + 1 is prime.
a(7) = 29, since 29 = 3^3 + 3^1 - 1 is prime.
a(8) = 31, since 31 = 3^3 + 3^1 + 1 is prime.
a(9) = 41.
a(10) = 43.
a(99) = 43889.
a(100) = 43891.
a(999) = 233524241.
a(1000) = 233524243.
a(9999) = 110211052379.
a(10000) = 110211052381.
a(99999) = 27208914574871.
a(100000) = 27208914574873.
a(199999) = 136140088764371.
a(200000) = 136140088764373.
[the last two terms form the 100000th twin prime pair of the form b^i + b^j +-1]

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

Cf. A239709, A239710, A239712 - A239720.

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

Original entry on oeis.org

3, 11, 29, 83, 89, 107, 251, 269, 809, 971, 2213, 2267, 6563, 6569, 6803, 8747, 19709, 19763, 20411, 59051, 65609, 177173, 183707, 531521, 538001, 590489, 1594331, 1594403, 1595051, 1596509, 4782971, 4782977, 4783697, 14348909, 14349149, 14526053, 14880347

Offset: 1

Hieronymus Fischer, Mar 28 2014

nonn

The base-3 representation of a term 3^i + 3^j - 1 has base-3 digital sum = 1 + 2*j == 1 (mod 2).

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, ...

			a(1) = 3, since 3 = 3^1 + 3^0 - 1 is prime.
a(5) = 89, since 89 = 3^4 + 3^2 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..131 [a(123) corrected by _Georg Fischer_, Dec 22 2024]

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).

select(isprime, [seq(seq(3^i+3^j-1, j=0..i-1), i=1..25)])[];  # Alois P. Heinz, Dec 22 2024

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 *)

A239713
"Answers the n-th term of A239713.
  Usage: n A239713
  Answer: a(n)"
  | a b i j k p q terms |
  terms := OrderedCollection new.
  k := 0.
  b := 3.
  p := b.
  i := 1.
  [k < self] whileTrue:
         [j := 0.
         q := 1.
         [j < i and: [k < self]] whileTrue:
                   [a := p + q - 1.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q.
                   j := j + 1].
         i := i + 1.
         p := b * p].
     ^terms at: self
[by Hieronymus Fischer, Apr 14 2014]
--------------------

A239713
"Version 2: Answers the n-th term of A239713.
  Uses distinctPowersOf: b from A018900
  Usage: n A239713
  Answer: a(n)”
  | a k n terms |
  terms := OrderedCollection new.
  n := 1.
  k := 0.
  [k < self] whileTrue:
         [(a:= (n distinctPowersOf: 3) - 1)
              isPrime ifTrue:    [k := k + 1.
                                 terms add: a].
              n := n + 1].
  ^terms at: self
[by Hieronymus Fischer, Apr 22 2014]
-----------

A239713
  "Version 3: Answer an array of the first n terms of A239713.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239713
  Answer: #(3 11 29 ... ) [a(1) ... a(n)]”
  ^self primesWhichAreDistinctPowersOf: 3 withOffset: -1
[by Hieronymus Fischer, Apr 22 2014]

A239714 Primes of the form m = 4^i + 4^j - 1, where i > j >= 0.

Original entry on oeis.org

19, 67, 79, 271, 1039, 1087, 1279, 4099, 4111, 4159, 5119, 16447, 20479, 65539, 65551, 65599, 81919, 262147, 262399, 263167, 266239, 1049599, 1114111, 1310719, 4194319, 4194559, 4195327, 16842751, 17825791, 67108879, 268435459, 268435711, 272629759, 1073741827, 1073741839, 1073758207

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

The base-4 representation of a term 4^i + 4^j - 1 has base-4 digital sum = 1 + 3*j == 1 (mod 3).
In base-4 representation the first terms are 103, 1003, 1033, 10033, 100033, 100333, 103333, 1000003, 1000033, 1000333, 1033333, 10000333, 10333333, 100000003, 100000033, 100000333, 103333333, 1000000003, 1000003333, 1000033333, ...
Numbers m which satisfy m = 4^i + 4^j + 1 are never primes, since the base-4 digital sum of m is 3, and thus, m is divisible by 3.

			a(1) = 19, since 19 = 4^2 + 4^1 - 1 is prime.
a(4) = 271, since 271 = 4^4 + 4^2 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..111

Cf. A234310.

A239714
  "Answer an array of the first n terms of A239714.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239714
  Answer: #(19 67 79 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 4 withOffset: -1

A239715 Primes of the form m = 5^i + 5^j - 1, where i > j >= 0.

Original entry on oeis.org

5, 29, 149, 15629, 15649, 15749, 16249, 18749, 391249, 393749, 1968749, 9765629, 9781249, 244140749, 244218749, 292968749, 30517968749, 152587890649, 152587891249, 152587893749, 152597656249, 152636718749, 3814697281249, 3814697656249, 19073486328749, 95367441406249

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

The base-5 representation of a term 5^i + 5^j - 1 has base-5 digital sum = 1 + 4*j == 1 (mod 4).
In base-5 representation the first terms are 10, 104, 1044, 1000004, 1000044, 1000444, 1004444, 1044444, 100004444, 100044444, 1000444444, 10000000004, 10000444444, ...
All terms after the first have the last digit 9, since 5^i == 5 (mod 10), and thus 5^i + 5^j == 0 (mod 10).
All terms which have i > j > 1 end with the last 2 digits …49, since 5^k == 25 (mod 100) for k > 1, and thus 5^i + 5^j == 50 (mod 100).
All terms which have i > j > 1 end with the last 3 digits ...249, ...649, or ...749, since 5^k == 125 (mod 1000) or 5^k == 625 (mod 1000) for k > 2, and thus 5^i + 5^j == 250 (mod 1000), or 5^i + 5^j == 650 (mod 1000), or 5^i + 5^j == 750 (mod 1000).
Numbers m = 5^i + 5^j - 1 with odd i and j are not terms. Example: 78249 = 5^7 + 5^3 - 1 = 3*26083.

			a(1) = 5, since 5 = 5^1 + 5^0 - 1 is prime.
a(3) = 149, since 149 = 5^3 + 5^2 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..50

A239715
  "Answers an array of the first n terms of A239715.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239715
  Answer: #(5 29 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 5 withOffset: -1

A239716 Primes of the form m = 6^i + 6^j - 1, where i > j >= 0.

Original entry on oeis.org

41, 251, 1301, 1511, 46691, 47951, 279941, 1679831, 10077911, 10124351, 60466181, 60466391, 60473951, 362797091, 362797271, 362843711, 2176782371, 2237248511, 13060694051, 13121160191, 78364164101, 78364164311, 78364171871

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

The base-6 representation of a term 6^i + 6^j - 1 has base-6 digital sum = 1 + 5*j == 1 (mod 5).

In base-6 representation the first terms are 105, 1055, 10005, 10555, 1000055, 1005555, 10000005, 100000555, 1000000555, 1000555555, 10000000005, 10000000555,

			a(1) = 41, since 41 = 6^2 + 6^1 - 1 is prime.
a(2) = 251, since 251 = 6^3 + 6^2 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..60

Select[Union[Flatten[Table[6^i+6^j-1,{i,20},{j,0,i-1}]]],PrimeQ] (* Harvey P. Dale, Oct 05 2024 *)

A239716
  "Answers an array of the first n terms of A239716.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239716
  Answer: #(41 241 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 6 withOffset: -1

A239717 Primes of the form m = 7^i + 7^j - 1, where i > j >= 0.

Original entry on oeis.org

7, 349, 19207, 117991, 120049, 823591, 5765143, 5882449, 6588343, 40353949, 282475591, 2017680349, 2259801991, 13841289601, 14123762449, 96894775207, 96929364013, 678223072897, 678223075249, 4747567274743, 5425784582791

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

The base-7 representation of a term 7^i + 7^j - 1 has base-7 digital sum = 1 + 6*j == 1 (mod 6).

Numbers m that satisfy m = 7^i + 7^j + 1 are never primes, since the base-7 digital sum of m is 3, and thus, m is divisible by 3.

			a(1) = 7, since 7 = 7^1 + 7^0 - 1 is prime.
a(2) = 349, since 349 = 7^3 + 7^1 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..40

Select[Flatten[Table[7^x+7^y-1,{x,0,20},{y,0,x-1}]],PrimeQ] (* Harvey P. Dale, Aug 13 2023 *)

A239717
  "Answers an array of the first n terms of A239717.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239717
  Answer: #(7 349 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 7 withOffset: -1

cp's OEIS Frontend

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

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Smalltalk

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A239720 Primes of the form m = 10^i + 10^j - 1, where i > j >= 0.

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Mathematica

Smalltalk

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

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Smalltalk

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A239719 Primes of the form m = 9^i + 9^j - 1, where i > j >= 0.

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A239711 Twin primes of the form m = b^i + b^j +- 1, where i > j > 0, b > 1.

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

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Maple

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Smalltalk

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A239714 Primes of the form m = 4^i + 4^j - 1, where i > j >= 0.

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A239715 Primes of the form m = 5^i + 5^j - 1, where i > j >= 0.

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