A098102 -id:A098102 - cp's OEIS frontend

A097743 Numbers of the form 3*2^(p - 1) - 1 where p is prime.

5, 11, 47, 191, 3071, 12287, 196607, 786431, 12582911, 805306367, 3221225471, 206158430207, 3298534883327, 13194139533311, 211106232532991, 13510798882111487, 864691128455135231, 3458764513820540927

Offset: 1

Parthasarathy Nambi, Sep 21 2004

nonn

			p=2, 2^(2-1) + (2^2 - 1) = 5;
p=3, 2^(3-1) + (2^3 - 1) = 11.

Iain Fox, Table of n, a(n) for n = 1..467

Cf. A000396, A098102, A055010.

Table[3*2^(Prime[n] - 1) - 1, {n, 18}] (* Robert G. Wilson v, Sep 24 2004 *)

a(n) = 3*2^(prime(n) - 1) - 1; \\ Michel Marcus, Nov 30 2017

a(n) = A001348(n) + A061286(n). - Iain Fox, Dec 08 2017

Edited and extended by Robert G. Wilson v, Sep 24 2004

A358527 Position of p in the factorization (without multiplicity) of 2^(p-1)-1, where p is the n-th odd prime.

Original entry on oeis.org

1, 2, 2, 2, 4, 3, 3, 2, 3, 4, 6, 6, 3, 2, 3, 2, 8, 4, 5, 8, 3, 2, 5, 6, 6, 3, 2, 8, 6, 6, 4, 4, 4, 3, 5, 7, 5, 2, 3, 2, 14, 4, 7, 7, 8, 9, 3, 2, 5, 5, 4, 12, 4, 4, 2, 3, 8, 7, 12, 3, 3, 6, 4, 10, 3, 9, 13, 2, 7, 7, 2, 3, 5, 8, 2, 3, 13, 10, 10, 4, 19, 4, 13, 3

Offset: 1

G. L. Honaker, Jr., Nov 20 2022

nonn

			a(19) = 5 because the 19th odd prime is 71 and 71 is the 5th largest distinct prime factor of 2^(71-1)-1 = 1180591620717411303423 = 3 * 11 * 31 * 43 * 71 * 127 * 281 * 86171 * 122921.

Amiram Eldar, Table of n, a(n) for n = 1..196

Cf. A065091, A098102, A358699.

Array[FirstPosition[FactorInteger[2^(# - 1) - 1], #][[1]] &[Prime[# + 1]] &, 50] (* Michael De Vlieger, Nov 27 2022 *)

a(n) = my(p=prime(n+1), v=factor(2^(p-1)-1)[,1]); vecsearch(v, p); \\ Michel Marcus, Nov 28 2022

More terms from Amiram Eldar, Nov 23 2022

A358699 a(n) is the largest prime factor of 2^(prime(n) - 1) - 1.

Original entry on oeis.org

3, 5, 7, 31, 13, 257, 73, 683, 127, 331, 109, 61681, 5419, 2796203, 8191, 3033169, 1321, 599479, 122921, 38737, 22366891, 8831418697, 2931542417, 22253377, 268501, 131071, 28059810762433, 279073, 54410972897, 77158673929, 145295143558111, 2879347902817, 10052678938039

Offset: 2

Hugo Pfoertner, Nov 27 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 2..197 (terms 2..120 from Hugo Pfoertner)

Cf. A005420, A006093, A006530, A061286, A071243, A086019, A098102.

Subsequence of A005420 and of A274906.

forprime (p=3, 140, my(f=factor(2^(p-1)-1)); print1(f[#f[,1],1],", "))

from sympy import primefactors, sieve
def A358699(n): return primefactors(2**(sieve[n]-1)-1)[-1] # Karl-Heinz Hofmann, Nov 28 2022

a(n) = A006530(A098102(n)). - Michel Marcus, Nov 28 2022

a(n) = A005420(A006093(n)). - Amiram Eldar, Dec 01 2022

A279882 a(n) = 2^(prime(n) + 1) - 1.

Original entry on oeis.org

7, 15, 63, 255, 4095, 16383, 262143, 1048575, 16777215, 1073741823, 4294967295, 274877906943, 4398046511103, 17592186044415, 281474976710655, 18014398509481983, 1152921504606846975, 4611686018427387903, 295147905179352825855, 4722366482869645213695

Offset: 1

Jaroslav Krizek, Dec 21 2016

Numbers whose binary representation is 1 repeated (prime(n)+1) times.

The only prime term is 7.

			For n=3; a(3) = 2^(prime(3) + 1) - 1 = 2^(5 + 1) - 1 = 2^6 - 1 = 63.

Cf. A101304 (2^(prime(n)+1)+1), A098102 (2^(prime(n)-1)-1), A278741 (2^(prime(n)-1)+1).

Magma
```
[2^(NthPrime(n)+1)-1: n in[1..50]]
```

A279882:=n->2^(ithprime(n)+1)-1: seq(A279882(n), n=1..30); # Wesley Ivan Hurt, Jan 23 2017

f[n_] := 2^(Prime[n]+1)-2; Array[f, 20] (* Robert G. Wilson v, Dec 21 2016 *)
2^(Prime[Range[20]]+1)-1 (* Harvey P. Dale, Jul 29 2024 *)

a(n) = 2^(prime(n)+1)-1 \\ Felix Fröhlich, Dec 21 2016

a(n) = A101304(n) - 2.

a(n) = A000225(A008864(n)). - Felix Fröhlich, Dec 21 2016

A359395 Least odd prime p in position n in the prime factorization of M(p) = 2^(p - 1) - 1.

Original entry on oeis.org

3, 5, 17, 13, 71, 37, 157, 61, 211, 313, 1289, 241, 337, 181, 577, 601, 541, 1381, 421, 1201, 1009, 1621, 1873, 3433, 4561, 1801, 3301, 2161, 3061, 5281, 3361, 2521, 7393, 6481, 4201, 4621, 8737, 9181, 6301, 19501, 7561, 16633, 12241, 26881, 15601, 9241, 21001, 14281, 12601, 53551

Offset: 1

Jean-Marc Rebert, Dec 31 2022

nonn

			M(5) = 15 = 3*5 and 5 is in second position in the prime factorization of M(5), and no lesser odd prime satisfies this, so a(2) = 5.
M(17) = 65535 = 3*5*17*257 and 17 is in third position in the prime factorization of M(17), and no lesser odd prime satisfies this, so a(3) = 17.

Carlos Rivera, Puzzle 1116. A358527, The Prime Puzzles & Problems Connection.

Cf. A098102, A358527.

f[n_] := Module[{p = Prime[n]}, Count[Prime[Range[n - 1]], ?(PowerMod[2, p - 1, #] == 1 &)] + 1]; f[1] = 0; With[{s = Array[f, 4000]}, ind = TakeWhile[ FirstPosition[s, #] & /@ Range[Max[s]] // Flatten, NumberQ ]; Prime[ind]] (* _Amiram Eldar, Jan 02 2023 *)

isok(p, n) = my(f=factor(2^(p-1) - 1, p+1)); if (#f~ < n, 0, f[n,1] == p);
a(n) = my(p=3); while(!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Jan 13 2023

cp's OEIS Frontend

A097743 Numbers of the form 3*2^(p - 1) - 1 where p is prime.

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A358527 Position of p in the factorization (without multiplicity) of 2^(p-1)-1, where p is the n-th odd prime.

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A358699 a(n) is the largest prime factor of 2^(prime(n) - 1) - 1.

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A279882 a(n) = 2^(prime(n) + 1) - 1.

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A359395 Least odd prime p in position n in the prime factorization of M(p) = 2^(p - 1) - 1.

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