A258091 -id:A258091 - cp's OEIS frontend

A258073 a(n) = 1 + 78557*2^n.

157115, 314229, 628457, 1256913, 2513825, 5027649, 10055297, 20110593, 40221185, 80442369, 160884737, 321769473, 643538945, 1287077889, 2574155777, 5148311553, 10296623105, 20593246209, 41186492417, 82372984833, 164745969665, 329491939329

Offset: 1

Tom Edgar, May 18 2015

78557 is the (conjectured) smallest Sierpiński number (A076336). This means that every number in the current sequence is composite.

Every number in the sequence is divisible by some number in {3, 5, 7, 13, 19, 37, 73}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
W. Sierpiński, Sur un problème concernant les nombres k * 2^n + 1, Elem. Math., 15 (1960), pp. 73-74.
Wikipedia, Sierpinski Number
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A076336.

Cf. A258091 (smallest prime factors).

a258073 = (+ 1) . (* 78557) . (2 ^)  -- Reinhard Zumkeller, May 19 2015

[1+78557*2^n: n in [1..25]]; // Vincenzo Librandi May 19 2015

Table[1 + 78557 2^n, {n, 1, 25}] (* Vincenzo Librandi, May 19 2015 *)

Sage
```
[78557*2^n+1 for n in [1..25]]
```

G.f.: x*(157115-157116*x)/((1-2*x)*(1-x)). - Vincenzo Librandi, May 19 2015

a(n) = 3*a(n-1)-2*a(n-2). - Wesley Ivan Hurt, Apr 26 2021

A258095 Numbers m, such that the smallest prime factor of 1+78557*2^m doesn't belong to the covering set {3, 5, 7, 13, 19, 37, 73}.

Original entry on oeis.org

39, 183, 219, 1047, 1227, 1299, 1875, 2271, 2559, 2703, 3315, 3531, 3819, 4359, 5079, 5187, 5403, 6015, 6339, 6447, 6843, 7491, 7599, 7671, 8499, 8535, 8859, 9327, 9579, 10119, 10155, 10623, 10983, 11379, 11667, 11811, 12639, 12711, 13467, 13755, 13899

Offset: 1

Reinhard Zumkeller, May 19 2015

nonn

A258091(a(n)) < 73, as each term in A258073 has at least one prime factor in the covering set.

			a(1) = 39; A258073(39) = 43187167471599617 = 71 * 73 * 211 * 39490356709, and 71 is not an element of the covering set.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Wikipedia, Sierpinski Number
Wikipedia, Covering set

Cf. A258073, A258091, A020639.

a258095 n = a258095_list !! (n-1)
a258095_list = filter
               (\x -> a258091 x `notElem` [3, 5, 7, 13, 19, 37, 73]) [1..]

A336347 Least prime factor of 44745755^4*2^(4n+2) + 1.

Original entry on oeis.org

13, 101, 29, 13, 39877, 41, 13, 37, 18661, 13, 41, 73, 13, 5719237, 144341, 13, 29, 89, 13, 353, 41, 13, 64450569241, 29, 13, 37, 101, 13, 89, 53, 13, 113, 313, 13, 37, 41, 13, 29, 73, 13, 41, 181, 13, 37, 29, 13, 857, 73, 13, 389, 41, 13, 37

Offset: 0

Jeppe Stig Nielsen, Jul 19 2020

nonn

There are k such that k*2^m + 1 is not prime for any m (then k is called a Sierpiński number). Erdős once conjectured that for such a k, the smallest prime factor of k*2^m + 1 would be bounded as m tends to infinitiy. The proven Sierpiński number k=44745755^4 is thought to be the first counterexample to this conjecture.
This sequence is either unbounded (in which case 44745755^4 is in fact a counterexample) or periodic.
a(229) <= 3034663491871541. - Chai Wah Wu, Aug 09 2020

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..228
M. Filaseta et al., On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, Journal of Number Theory, Volume 128, Issue 7 (July 2008), pp. 1916-1940.
Anatoly S. Izotov, A Note on Sierpinski Numbers, Fibonacci Quarterly (1995), pp. 206-207.

Cf. A020639, A076336, A213353, A258091, A336943.

A279798 Largest prime factor of 78557*2^n + 1.

Original entry on oeis.org

67, 104743, 8609, 281, 521, 1163, 1436471, 12391, 136343, 1483, 23663, 727, 10453, 2029, 135481883, 7429021, 2059324621, 6864415403, 3716857, 9629, 451358821, 51782483, 62504399, 439322585771, 63337, 128110399, 42209, 59569669, 111486983, 10936129, 31585821557

Offset: 1

Seiichi Manyama, Dec 19 2016

nonn

a(n) <= A258073(n) / A258091(n).

			78557 * 2^1 + 1 = 157115 = 5 * 7 * 67^2. So a(1) = 67.
78557 * 2^2 + 1 = 314229 = 3 * 104743. So a(2) = 104743.

Daniel Suteu, Table of n, a(n) for n = 1..412
Wikipedia, Sierpinski Number

Cf. A002587, A258073, A258091.

Table[FactorInteger[2^n * 78557 + 1][[-1, 1]], {n, 30}] (* Alonso del Arte, Jan 01 2017 *)

a(n) = my(k=78557*2^n+1); factor(k)[omega(k), 1] \\ Felix Fröhlich, Jan 01 2017

a(n) = A006530(A258073(n)).

cp's OEIS Frontend

A258073 a(n) = 1 + 78557*2^n.

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A258095 Numbers m, such that the smallest prime factor of 1+78557*2^m doesn't belong to the covering set {3, 5, 7, 13, 19, 37, 73}.

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A336347 Least prime factor of 44745755^4*2^(4n+2) + 1.

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A279798 Largest prime factor of 78557*2^n + 1.

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