A076336 -id:A076336 - cp's OEIS frontend

A137715 Prime values of n for which n*2^k + 1 is composite for all positive integers k.

271129, 322523, 327739, 482719, 934909, 1639459, 2131043, 2131099, 2576089, 3098059, 3608251, 4573999, 6678713, 6799831, 7523281, 7761437, 8184977, 8840599, 8879993, 8959163, 9208337, 9252323, 9930469, 9937637, 10192733, 10306187, 10391933, 11206501

Offset: 1

Ant King, Feb 09 2008

nonn

The sequence contains those members of A076336 that are prime.

Note that the terms in A076336 are presently conjectural. - Joerg Arndt, Jun 29 2015

			As 271129 is the first known prime value of n for which n*2^k + 1 is composite for all positive integers k, a(1) = 271129.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..3670
R. Baillie, G. Cormack, and H. C. Williams, The Problem of Sierpinski Concerning k*2^n+1, Mathematics of Computation, Vol. 37, No. 155 (July 1981), pp. 229-231. Corrigenda; Mathematics of Computation, Vol. 39, No. 159 (July 1982), p. 308.
Wilfrid Keller, Factors of Fermat Numbers and Large Primes of the Form k*2^n+1, Mathematics of Computation, Vol. 41, No. 164 (October 1983), pp. 661-673.
Mersenne Forum, The Prime Sierpinski Problem.
Seventeen or Bust, A Distributed Attack on the Sierpinski problem
W. Sierpinski, Sur un problème concernant les nombres k*2^n+1, Elem. d. Math. 15, pp. 73-74, 1960.

Cf. A057192, A076336, A094076.

More terms from Arkadiusz Wesolowski, Apr 24 2012

A187716 Odd numbers m divisible by 3 such that for every k >= 1, m*2^k + 1 has a divisor in the set {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 109, 151, 241, 331}.

Original entry on oeis.org

21484572547591559649, 50166404682516122859, 51814002736113272553, 53246606581410442023, 58992081042572747991, 65634687179877002283, 80269357428943941837, 92027572854849003627, 103083799330841020677

Offset: 1

Arkadiusz Wesolowski, Mar 17 2011

nonn

Wilfrid Keller (2004, published) gave the first known example.
21484572547591559649 computed in 2017 by the author.
Conjecture: 21484572547591559649 is the smallest Sierpiński number that is divisible by 3. - Arkadiusz Wesolowski, May 12 2017
The above conjecture is false, because the Sierpiński number 7592506760633776533 is a counterexample. - Arkadiusz Wesolowski, Jul 27 2023

Chris Caldwell, The Prime Glossary, Sierpinski number
Carlos Rivera, Problem 49. Sierpinski-like numbers, The Prime Puzzles & Problems Connection.

Cf. A076336, A187714.

Name changed and entry revised by Arkadiusz Wesolowski, May 11 2017

A213353 A subset of numbers n such that n^4 is a Sierpinski number.

Original entry on oeis.org

44745755, 1812338107, 9266824499, 12308871853, 13657352875, 22767480811, 22930161667, 24068927659, 25549554505, 25770503549, 57939582163, 90219135299, 90329609821, 96949951147, 103126759951

Offset: 1

Arkadiusz Wesolowski, Jun 09 2012

A sequence constructed from Izotov's trick.

If n belongs to this sequence and n does not end in 5, then n^4 has the covering set {3, 5, 17, 97, 241, 257, 673}.

Anatoly S. Izotov, A note on Sierpinski numbers
Wikipedia, Sierpinski number

Subset of A233469. Cf. A076336.

(* even if nn is increased, no additional terms are generated *) nn = 14; lst = {}; n = 44745755; p = 2^12; m = 3*(p^4 - 1)/(p - 1); Do[a = n + (-1)^c*m; n = a/GCD[a, p]; AppendTo[lst, Abs@n], {c, 0, nn}]; Union@lst

A244563 Odd integers n such that for every integer k>0, n*2^k+1 has a divisor in the set { 3, 5, 7, 13, 19, 37, 109 }.

Original entry on oeis.org

1290677, 4095859, 5841947, 7158107, 8959163, 9044629, 9252323, 9933857, 10306187, 11000303, 15598231, 16010419, 16625747, 16907749, 18068693, 19428919, 20189993, 23487497, 25614893, 26471633, 28410121, 30375901, 30666137, 32552687

Offset: 1

Pierre CAMI, Jun 30 2014

nonn

For n > 144 a(n) = a(n-144) + 209191710, the first 144 values are in the table.

Pierre CAMI, Table of n, a(n) for n = 1..144

Cf. A076336, A244072, A244561, A244562.

For n > 144 a(n) = a(n-144) + 209191710.

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

Original entry on oeis.org

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

A260350 Define g(k) = min(n: n >= 0, 2^n + k prime). Then a(n) = min(odd k: g(k) = n).

Original entry on oeis.org

1, 3, 7, 23, 31, 47, 199, 83, 61, 257, 139, 953, 991, 647, 1735, 383, 511, 1337, 1069, 713, 271, 1937, 3223, 5213, 751, 8477, 4339, 353, 1501, 287, 829, 1553, 2371, 1811, 11185, 3023, 7381, 7937, 6439, 1433, 13975, 2897, 4183

Offset: 0

Hugo van der Sanden, Jul 23 2015

nonn

Previous name: a(n) = min(k : A067760((k-1)/2)) = n.
a(n) is the first odd number k for which 2^m + k is the first prime value, as m ranges from 0 to n, or 0 if no such k exists. Thus it is the first k for which A067760((k-1)/2) = n, and therefore also the first k for which you need to test primality of exactly n values to show that it is not a dual Sierpiński number.
In the name, g(n) = A067760(n) except for n=1. - Michel Marcus, Apr 07 2018

			2^i + 7 is composite for i < 2 (with values 8, 9) but prime for i = 2 (11); the smaller odd numbers 1, 3 and 5 each yield a prime for smaller i, so a(2) = 7.

Hugo van der Sanden, Table of n, a(n) for n = 0..1087

Cf. A067760, A076336, A260349.

g(k) = {my(j=0); while (!isprime(2^j+k), j++); j;}
a(n) = {my(k = 1); while(g(k) != n, k+=2); k;} \\ Michel Marcus, Apr 07 2018

For n>=2, a(n) = (min(k : A067760((k-1)/2)) = n). - Michel Marcus, Apr 07 2018

New name from Hugo van der Sanden and Michel Marcus, Apr 07 2018

A263644 Odd numbers that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

Original entry on oeis.org

30666137, 31210219, 52109063, 52504261, 55414847, 55876981, 57816799, 60097043, 63723707, 68748319, 79933129, 87747827, 88486403, 93034073, 104218883, 131873509, 138385817, 152485283, 155269609, 158241023, 165795677, 166441831, 177702619, 197903207

Offset: 1

Arkadiusz Wesolowski, Oct 22 2015

nonn

Odd n such that for all k > 0 the numbers n + 2^k and n - 2^k are nonprimes.

Cf. A006285, A076335, A076336. Subsequence of A255967. A263645 gives the primes.

A006285 INTERSECT A076336.

A364413 Odd numbers m such that for every k >= 1, m*2^k + 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

Original entry on oeis.org

189035277393779, 212050850472529, 618127765127603, 777947701660121, 1171304921532749, 1358735367828947, 1834310020939021, 2357654372323739, 2638037471052913, 3025664372930897, 3935005074246167, 4688754513654559, 4996748200142999, 5425272498782051, 5455203077891285

Offset: 1

Arkadiusz Wesolowski, Jul 23 2023

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34560

Cf. A076336, A076335, A180247, A291360, A364412.

For n > 34560, a(n) = a(n-34560) + 10014447295554878022.

A373801 a(1) = 2; thereafter, if a(n-1) is prime then a(n) = prime(n) + 1; otherwise a(n) = 2*a(n-1) - 1.

Original entry on oeis.org

2, 4, 7, 8, 15, 29, 18, 35, 69, 137, 32, 63, 125, 249, 497, 993, 1985, 3969, 7937, 72, 143, 285, 569, 90, 179, 102, 203, 405, 809, 114, 227, 132, 263, 140, 279, 557, 158, 315, 629, 1257, 2513, 5025, 10049, 20097, 40193, 200, 399, 797, 228, 455, 909, 1817, 3633, 7265, 14529, 29057, 58113, 116225

Offset: 1

N. J. A. Sloane, Aug 05 2024

nonn

Inspired by A374965. Just as the Riesel numbers (A101036 etc.) underlie A374965, so the Sierpinski numbers (A076336 etc.) underlie the present sequence. This means that for both A374965 and the present sequence, it is possible that there are only finitely many prime terms.
What is the next prime after a(1336) = 1486047139543908353?
The next prime in the sequence after a(1336) is the 328-digit prime a(2412) = 11027*2^1075 + 1 =
44637792944394283771459323765390022896709223538983902782431025499369487088325693\
80355294302151494343616855815219642969893790841894306289338825113522293047097809\
14527499539453353195318334412379318970183638103791974206651303944817277532365140\
54865648555402249863235603037071611259242935028448372668756790221309881865220759\
33966337. - Alois P. Heinz, Aug 05 2024
For a(1) any prime, the trajectory converges to this sequence. Just as in A374965, the trajectories appear to converge to a few attractors. In fact it appears that for most values of a(1), the trajectory converges to the present sequence. However, for a(1) = 384 and 767 the trajectories are different. - Chai Wah Wu, Aug 07 2024

N. J. A. Sloane, Table of n, a(n) for n = 1..2000
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A374965, A076336, A101036.

For the primes in this sequence, see A373802 and A373803.

A:=Array(1..1200,0);
t:=2;
A[1]:= t;
for n from 2 to 100 do
if isprime(t) then t:=ithprime(n)+1; else t:=2*t-1; fi;
A[n]:=t;
od:
[seq(A[n],n=1..100)];

Module[{n = 1}, NestList[If[n++; PrimeQ[#], Prime[n] + 1, 2*# - 1] &, 2, 100]] (* Paolo Xausa, Aug 07 2024 *)

from sympy import isprime, nextprime
def A373801_gen(): # generator of terms
    a, p = 2, 3
    while True:
        yield a
        a, p = p+1 if isprime(a) else (a<<1)-1, nextprime(p)
A373801_list = list(islice(A373801_gen(),20)) # Chai Wah Wu, Aug 05 2024

A133831 Least positive number k != n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 9, 3, 3, 2, 1, 4, 5, 1, 1, 11, 1, 6, 5, 4, 7, 3, 9, 27, 17, 15, 1, 15, 1, 6, 458465, 4, 9, 14, 13, 3, 11, 25, 57, 6, 7, 46, 17, 7, 15, 2, 1009, 30, 23, 6, 21, 2, 33, 1, 1265, 3, 69, 14, 5, 6, 21, 19, 2241, 30, 3, 1, 5, 34, 19, 26, 17, 19, 17, 5, 33, 15, 23, 27

Offset: 1

T. D. Noe, Sep 26 2007

nonn

Does such k exist (so that a(n) is nonzero) for all n? These binary trinomials can also be written as f*2^n+1, where f=2^m+1 for some m, which is reminiscent of the Sierpinski problem (see A076336). Hence if there are no Sierpinski numbers of the form 2^m+1, then a(n) is nonzero for all n.

The PFGW program was used to find a(32), which produces a 138012-digit probable prime. If a(256) is nonzero, it is greater than 10^6.

T. D. Noe, Table of n, a(n) for n = 1..255
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+2^m+1, PRP Top Records.

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).
Closely related problems: A040076 (see also A076336), A067760, A133830 (k < n), A133832 (k > n).
Cf. A095056.

mx=4000; Table[s=1+2^n; k=1; While[k==n || (k

Edited by Peter Munn, Sep 29 2024

cp's OEIS Frontend

A137715 Prime values of n for which n*2^k + 1 is composite for all positive integers k.

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A187716 Odd numbers m divisible by 3 such that for every k >= 1, m*2^k + 1 has a divisor in the set {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 109, 151, 241, 331}.

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A213353 A subset of numbers n such that n^4 is a Sierpinski number.

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A244563 Odd integers n such that for every integer k>0, n*2^k+1 has a divisor in the set { 3, 5, 7, 13, 19, 37, 109 }.

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

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A260350 Define g(k) = min(n: n >= 0, 2^n + k prime). Then a(n) = min(odd k: g(k) = n).

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A263644 Odd numbers that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

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A364413 Odd numbers m such that for every k >= 1, m*2^k + 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

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A373801 a(1) = 2; thereafter, if a(n-1) is prime then a(n) = prime(n) + 1; otherwise a(n) = 2*a(n-1) - 1.

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