A059054 -id:A059054 - cp's OEIS frontend

A059055 Primes which can be written as (b^k+1)/(b+1) for positive integers b and k.

3, 7, 11, 13, 31, 43, 61, 73, 157, 211, 241, 307, 421, 463, 521, 547, 601, 683, 757, 1123, 1483, 1723, 2551, 2731, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9091, 9901, 10303, 11131, 12211, 12433, 13421, 13807, 14281

Offset: 1

Henry Bottomley, Dec 21 2000

For (b^k+1)/(b+1) to be a prime, k must be an odd prime. 2=(0^0+1)/(0+1) has been excluded since neither b nor k would be positive.
From Bernard Schott, Apr 30 2021: (Start)
43 is the only known prime to have two such representations (examples).
The next two sequences realize a partition of this set: Brazilian primes of the form (c^q-1)/(c-1) (A002383 \ {3}) and primes that are not Brazilian (A343774). (End)

			43 is in the sequence since (2^7+1)/(2+1) = 129/3 = 43; indeed also (7^3+1)/(7+1) = 344/8 = 43.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 3880 terms from T. D. Noe)
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.

Cf. A002383, A059054.

Cf. A003424, A085104.

max = 89; maxdata = (1 + max^3)/(1 + max); a = {}; Do[i = 1; While[i = i + 2; cc = (1 + m^i)/(1 + m); cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a] (* Lei Zhou, Feb 08 2012 *)

isok(p) = {if (isprime(p), for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1););););} \\ Michel Marcus, Apr 30 2021

A206944 Numbers Phi_k(m) with integer k > 2, |m| > 1 but k != 2^j (j > 1).

Original entry on oeis.org

3, 7, 11, 13, 21, 31, 43, 57, 61, 73, 91, 111, 121, 127, 133, 151, 157, 183, 205, 211, 241, 273, 307, 331, 341, 343, 381, 421, 463, 507, 521, 547, 553, 601, 651, 683, 703, 757, 781, 813, 871, 931, 993, 1057, 1093, 1111, 1123, 1191, 1261, 1333, 1407, 1483

Offset: 1

Lei Zhou, Feb 13 2012

nonn

Phi_k(m) denotes the cyclotomic polynomial numbers Cyclotomic(k,m).
There is a property for Cyclotomic(k,m):
Cyclotomic(k^(j+1),m) = Cyclotomic(k,m^(k^j)).
So actually when k=2^(j+1), j is a positive integer,
Cyclotomic(k,m) = Cyclotomic(2,m^(2^j)) = 1+m^(2^j).
If these cases are excluded from A206942, this sequence is obtained.
This sequence is a subsequence of A206942.
Sequence A059054 is a subsequence of this sequence.
The Mathematica program can generate this sequence to arbitrary boundary maxdata without a user's choice of parameters. The parameter determination part of this program is explained at A206864.

			a(1) = 3 = Phi(6,2).
5 = Phi(4,2) = Phi(2,4) so excluded.
a(2) = 7 = Phi(3,2).

Cf. A206942, A194712, A059054, A206864.

phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; maxdata = 1500; max = Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/4]*2; eb = 2*Floor[(Log[2, maxdata])/2 + 0.5]; While[eg = phiinv[eb]; lu = Length[eg]; lu == 0, eb = eb + 2]; t = Select[Range[eg[[Length[eg]]]], ((EulerPhi[#] <= eb) && ((! IntegerQ[Log[2, #]]) || (# <= 2))) &]; ap = SortBy[t, Cyclotomic[#, 2] &]; an = SortBy[t, Cyclotomic[#, -2] &]; a = {}; Do[i = 2; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata,
  a = Append[a, cc]]; i = 2; While[i++; cc = Cyclotomic[an[[i]], -m]; cc <= maxdata, a = Append[a, cc]], {m, 2, max}]; Union[a]

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A059055 Primes which can be written as (b^k+1)/(b+1) for positive integers b and k.

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