A239719 -id:A239719 - cp's OEIS frontend

A242482 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 1.

2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 42, 43, 45, 47, 49, 51, 53, 54, 55, 56, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 97, 99

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

Numbers n such that A242480(n) = (1/2*n*(n+1)) mod n + sigma(n) mod n + antisigma(n) mod n = (A142150(n) + A054024(n) + A229110(n)) = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) = n. Numbers n such that A242481(n) = (A142150(n) + A054024(n) + A229110(n)) / n = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) / n = 1.
Conjecture: with number 1 complement of A242483.
Supersequence of primes (A000040).
If there is no odd multiply-perfect number, then:
(1) a(n) = union of odd numbers >= 3 and even numbers from A239719.
(2) a(n) = supersequence of odd numbers (A005408).

			6 is in sequence because [(6*(6+1)/2) mod 6 + sigma(6) mod 6 + antisigma(6) mod 6] / 6 = (21 mod 6 + 12 mod 6 + 9 mod 6) / 6 = (3 + 0 + 3 ) / 6 = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..5000

Cf. A242480, A242481, A242483, A242484, A242485, A242486.

[n: n in [1..1000] | n eq ((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n))]

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]

A239718 Primes of the form m = 8^i + 8^j - 1, where i > j >= 0.

Original entry on oeis.org

71, 4159, 32831, 262151, 266239, 294911, 2101247, 18874367, 134479871, 1073741831, 68721573887, 549755813951, 4398046515199, 4398046543871, 4398046773247, 4398063288319, 281474976711167, 281474976743423, 281474978807807, 281474993487871, 282024732524543

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

The base-8 representation of a term 8^i + 8^j - 1 has base-8 digital sum = 1 + 7*j == 1 (mod 7).
In base-8 representation the first terms are 107, 10077, 100077, 1000007, 1007777, 1077777, 10007777, 107777777, 1000777777, 10000000007, 1000007777777, 10000000000077, 100000000007777, ...
Numbers m that satisfy m = 8^i + 8^j - 1 with odd i and j are not terms. Example: 33279 = 8^5 + 8^3 - 1 = 3*11093.

			a(1) = 71, since 71 = 8^2 + 8^1 - 1 is prime.
a(2) = 4159, since 4159 = 8^4 + 8^2 - 1 is prime.

Georg Fischer, Table of n, a(n) for n = 1..40 [First 35 terms from Hieronymus Fischer]

Cf. A018900, A239709, A239712 (base 2), A239713 (base 3), A239714 (base 4), A239715 (base 5), A239716 (base 6), A239717 (base 7), A239719 (base 9), A239720 (base 10).

select(isprime, [seq(seq(8^i+8^j-1, j=0..i-1), i=1..25)])[];  # Alois P. Heinz, Dec 22 2024

A239718
  "Answers an array of the first n terms of A239718.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
Usage: n A239718
Answer: #(71 4159 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 8 withOffset: -1

cp's OEIS Frontend

A242482 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 1.

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

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A239718 Primes of the form m = 8^i + 8^j - 1, where i > j >= 0.

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Maple

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