A239712 -id:A239712 - cp's OEIS frontend

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

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

A128898 Primes of form 2^j + 2^k - 1 or 2^j + 2^k + 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 67, 71, 73, 79, 97, 127, 131, 137, 191, 193, 257, 263, 271, 383, 521, 577, 641, 769, 1031, 1033, 1039, 1087, 1151, 1153, 1279, 2053, 2063, 2081, 2111, 2113, 4099, 4111, 4127, 4129, 4159, 5119, 6143, 8191, 8209

Offset: 1

J. M. Bergot, Apr 21 2007

nonn

Union of A000668, A081091 and A239712. - Robert Israel, Jun 13 2018

			2^2 + 2^5 + 1 = 4 + 32 + 1 = 37 is prime, hence 37 is a term.
2^4 + 2^5 - 1 = 16 + 32 - 1 = 47 is prime, hence 47 is a term.
2^3 + 2^6 + 1 = 8 + 64 + 1 = 73 is prime, hence 73 is a term.

Robert Israel, Table of n, a(n) for n = 1..6772

Cf. A000668 (Mersenne primes), A092506 (primes of form 2^n + 1), A070739 (primes of form 2^x+2^y+1), A081091, A239712.

sort(convert(select(isprime, {2,seq(seq(seq(2^i+2^j+k,k=[-1,1]),j=1..i),i=1..15)}),list)); # Robert Israel, Jun 13 2018

lst = {}; Do[p = 2^a + 2^b; If[PrimeQ[p - 1], AppendTo[lst, p - 1]]; If[PrimeQ[p + 1], AppendTo[lst, p + 1]], {a, 0, 14}, {b, 0, a}]; Union@ lst (* Robert G. Wilson v *)

{m=13; v=[]; for(j=0, m, for(k=j, m, if(isprime(p=2^j+2^k-1), v=concat(v, p)); if(isprime(p=2^j+2^k+1), v=concat(v,p)))); w=Vec(listsort(List(v), 1)); w} /* Klaus Brockhaus, Apr 22 2007 */

Edited, corrected and extended by Klaus Brockhaus and Robert G. Wilson v, Apr 22 2007

A264908 Primes of the form 2^i + 2^j + 2^k - 1, i > j > k > 0.

Original entry on oeis.org

13, 37, 41, 43, 73, 83, 97, 103, 137, 139, 151, 163, 167, 193, 199, 223, 521, 523, 547, 577, 607, 641, 643, 647, 769, 1033, 1063, 1091, 1103, 1153, 1283, 1543, 1567, 1663, 2053, 2081, 2083, 2087, 2113, 2143, 2179, 2207, 2239, 2311, 2591, 2687, 3079, 3583, 4129, 4231, 4639

Offset: 1

Robert G. Wilson v, Nov 28 2015

nonn

Cf. A000668, A095077, A239712.

Select[ Flatten[ Table[2^i + 2^j + 2^k - 1, {i, 3, 10}, {j, 2, i - 1}, {k, j - 1}]], PrimeQ]

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

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A128898 Primes of form 2^j + 2^k - 1 or 2^j + 2^k + 1.

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A264908 Primes of the form 2^i + 2^j + 2^k - 1, i > j > k > 0.

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Mathematica