A076336 -id:A076336 - cp's OEIS frontend

A033919 Odd k for which k+2^m is composite for all m < k.

773, 2131, 2491, 4471, 5101, 7013, 8543, 10711, 14717, 17659, 19081, 19249, 20273, 21661, 22193, 28433, 35461, 37967, 39079, 40291, 41693, 48527, 60443, 60451, 60947, 64133, 75353, 78557

Offset: 1

Dan Hoey

nonn

Related to the Sierpiński number problem.

In an archived website, Payam Samidoost gives these numbers and other results about the dual Sierpiński problem. It is conjectured that, for each of these k<78557, there is an m such that k+2^m is prime. Then a covering argument would show that 78557 is the least odd number such that 78557+2^m is composite for all m. The impediment in the "dual" problem is that it is currently very difficult to prove the primality of large numbers of the form k+2^m. It is much easier to prove the Proth primes of the form k*2^m+1 which occur in the usual Sierpiński problem. According to the distributed search project "Five or Bust", 40291 is the only value of k < 78557 for which there is currently no m known making k + 2^m a prime or probable prime. - T. D. Noe, Jun 14 2007 and Phil Moore (moorep(AT)lanecc.edu), Dec 14 2009

Mersenneforum, Five or Bust
Payam Samidoost, The dual Sierpinski problem search (Archive of the site at the Wayback Machine, original link is dead)
Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind.

Cf. A067760, A076336.

t={}; Do[k=1; While[kT. D. Noe, Jun 14 2007 *)

More terms from David W. Wilson
More terms from T. D. Noe, Jun 14 2007
Out-of-date information from Payam Samidoost's website corrected, using the current status on the dual Sierpiński problem from "Five or Bust," by Phil Moore (moorep(AT)lanecc.edu), Dec 14 2009
Broken link to Payam Samidoost's website replaced with link to archive in the Wayback Machine by Felix Fröhlich, Jul 11 2014
26213 removed from sequence following an email message from Maximilian Pacher, who reports that 2^1271+26213 is prime. - N. J. A. Sloane, Dec 31 2015

A206001 Least k such that k*2^m + 1 has a covering set with precisely n primes.

Original entry on oeis.org

271129, 78557, 1777613, 169073869

Offset: 6

Arkadiusz Wesolowski, Feb 10 2012

a(10) > 2^31. a(11) = 1143502909. - Arkadiusz Wesolowski, Apr 29 2012

			271129 has the covering set {3, 5, 7, 13, 17, 241}.
78557 has the covering set {3, 5, 7, 13, 19, 37, 73}.
1777613 has the covering set {3, 5, 7, 13, 17, 19, 109, 433}.
169073869 has the covering set {3, 5, 7, 11, 13, 41, 61, 151, 331}.

Wikipedia, Covering set
Wikipedia, Sierpinski number

Cf. A076336.

A244566 Odd integers n such that for every integer k>0, n*2^k+1 has a divisor in the set { 3, 5, 7, 13, 17, 97, 257 }.

Original entry on oeis.org

327739, 5455789, 8879993, 9043831, 21823667, 25763447, 29949559, 75037639, 92732027, 119863547, 119879899, 122091961, 146880319, 151060223, 152106751, 163378771, 181339441, 182384417, 182646049, 218039041, 232190537

Offset: 1

Pierre CAMI, Jun 30 2014

These are the Sierpiński numbers (A076336) with covering set {3, 5, 7, 13, 17, 97, 257}. - David W. Wilson, Jul 18 2014

For n > 96, a(n) = a(n-96) + 1156954890, the first 96 values are in the table.

Pierre CAMI, Table of n, a(n) for n = 1..96

Cf. A076336, A244076, A244561, A244562, A244563, A244564, A244565.

is(n)=my(G=578477445,t=Mod(n,G)); for(k=1,768,t*=2; if(gcd(t+1, G)==1, return(0))); n%2 \\ Charles R Greathouse IV, Jul 18 2014

For n > 96, a(n) = a(n-96) + 1156954890.

A263169 Smallest positive k such that 78557^k + 2^n is prime, or 0 if no such value exists.

Original entry on oeis.org

0, 33, 2, 0, 76

Offset: 0

Arkadiusz Wesolowski, Oct 11 2015

{0, 2, 3} is the set of all noncomposite numbers that belong to this sequence.

The sequence continues: a(5) = ?, a(6) = ?, 87, 12, 0, 0, 3, a(12) = ?.

Cf. A076336.

lst:=[]; for n in [1..4] do if not n mod 6 eq 3 then k:=2; while not IsPrime(78557^k+2^n) do k+:=1; end while; Append(~lst, k); else Append(~lst, 0); end if; end for; [0] cat lst;

A263391 Sierpiński numbers that have at least two covering sets.

Original entry on oeis.org

12151397, 31210219, 45181667, 56191673, 66887071, 68468753, 69169397, 71307347, 114921271, 122311103, 133228283, 152252267, 154337567, 182479909, 185282537, 192413177, 210465533, 220192013, 226521259, 235663343, 236281883, 253282909, 275248343, 282777829

Offset: 1

Arkadiusz Wesolowski, Oct 16 2015

nonn

			For every k >= 1, 12151397*2^k + 1 has a divisor in the set {3, 5, 7, 13, 19, 37, 73} and also in the set {3, 5, 7, 13, 19, 73, 109}. 12151397 is therefore in the sequence.

Wikipedia, Sierpinski number

Cf. A076336, A263392.

2 terms inserted by Arkadiusz Wesolowski, Aug 28 2016

More terms from Arkadiusz Wesolowski, Jan 09 2018

A269326 Let k be a number which is simultaneously Sierpiński and Riesel, and let P be a set of primes which cover every number of the form k2^m + 1 and of the form k2^m - 1 with m >= 1. Sequence shows elements of the set P which has the property that the product of its primes is as small as it is possible.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331

Offset: 1

Arkadiusz Wesolowski, Feb 23 2016

Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975), pp. 79-81.

Cf. A076335, A076336, A101036.

PrimeDivisors((2^36-1)*(2^48-1)*(2^60-1))[1..18];

A270271 Odd numbers n such that for every k >= 1, n*2^k + 1 has a divisor in the set {3, 5, 17, 257, 641, 65537, 6700417}.

Original entry on oeis.org

201446503145165177, 1007236913771681629, 1697906240793858917, 2331023822106839599, 2935363331541925531, 3367034409844073483, 3914042604075779837, 4863495246870308311, 5036162578625852633, 5590196669446332863, 6705290764721718679, 7284449444083822547

Offset: 1

Arkadiusz Wesolowski, Mar 14 2016

nonn

M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, pp. 72-73.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..64
Chris Caldwell, The Prime Glossary, Sierpinski number
W. Sierpiński, Sur un problème concernant les nombres k * 2^n + 1, Elem. Math., 15 (1960), pp. 73-74.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A257647. Subsequence of A076336.

lst:=[]; e:=2^64; P:=PrimeDivisors(e-1); C:=[1, 1, 1, 1, 1, 1, 33]; Pr:=&*[P[i]: i in [1..#P]]; S:=CRT([Modexp(2, C[i], P[i]): i in [1..#C]], P); for t in [1..33] do a:=S+Pr; g:=Gcd(a, e); S:=Floor(a/g); Append(~lst, S); end for; Sort(lst)[1..12];

a(n) = a(n-64) + 2*(2^64-1) for n > 64.

A270895 Sierpiński numbers that are squares.

Original entry on oeis.org

18213931681, 5044241991721, 6093734165401, 6278567569849, 17919144076201, 52536446729209, 56470430531761, 72659547739249, 107737943736721, 123343213788001, 136925780937841, 170306127919321, 211292941174561, 296693149397089, 321058916825689

Offset: 1

Arkadiusz Wesolowski, Mar 25 2016

This sequence contains infinitely many numbers that end in 5. See also A233469.

Wikipedia, Sierpinski number

Cf. A076336, A233469, A270896.

A000290 INTERSECT A076336.

a(n) = A270896(n)^2.

A270896 Values of n for which n^2 is a Sierpiński number.

Original entry on oeis.org

134959, 2245939, 2468549, 2505707, 4233101, 7248203, 7514681, 8524057, 10379689, 11105999, 11701529, 13050139, 14535919, 17224783, 17918117, 20887471, 22609447, 23352049, 26616221, 29156453, 29583953, 29913997, 31273301, 32046613, 35672821, 35882309

Offset: 1

Arkadiusz Wesolowski, Mar 25 2016

nonn

Wikipedia, Sierpinski number

Cf. A076336, A233469, A270895.

a(n) = sqrt(A270895(n)).

A270994 a(n) = 9454129 + 11184810*n.

Original entry on oeis.org

9454129, 20638939, 31823749, 43008559, 54193369, 65378179, 76562989, 87747799, 98932609, 110117419, 121302229, 132487039, 143671849, 154856659, 166041469, 177226279, 188411089, 199595899, 210780709, 221965519, 233150329, 244335139, 255519949, 266704759, 277889569, 289074379, 300259189

Offset: 0

Altug Alkan, Mar 28 2016

See A270971 for the motivation.
These are all Sierpiński numbers.
Since 9454129 is a term of A244561, for every integer k > 0, 9454129*2^k + 1 has a divisor in the set {3, 5, 7, 13, 17, 241}. And because 11184810 = 2*3*5*7*13*17*241, a(n)*2^k + 1 = 9454129*2^k + 1 + 11184810*n*2^k + 1 always has a divisor in the set {3, 5, 7, 13, 17, 241}. Since a(n) is always odd because of its definition, a(n) is a Sierpiński number.
Also 9454129 + 28 = 9454157 is a term of A244561. So, with the same proof, a(n) + 28 is a Sierpiński number too.
Are a(n) and a(n) + 28 always consecutive Sierpiński numbers?

			a(1) = 9454129 + 11184810*1 = 20638939.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A076336, A244561, A270971, A270993.

[9454129 + 11184810*n: n in [0..30]]; // Vincenzo Librandi, Mar 29 2016

A270994:=n->9454129 + 11184810*n: seq(A270994(n), n=0..40); # Wesley Ivan Hurt, Apr 02 2016

Table[9454129 + 11184810*n, {n, 0, 100}] (* G. C. Greubel, Mar 28 2016 *)

PARI
```
a(n) = 9454129 + 11184810*n;
```

x='x+O('x^99); Vec((9454129+1730681*x)/(1-x)^2)

for n in range(0,100):print(9454129+11184810*n) # Soumil Mandal, Apr 03 2016

G.f.: (9454129 + 1730681*x)/(1 - x)^2.

a(n) = 2*a(n-1) - a(n-2) for n > 1.

cp's OEIS Frontend

A033919 Odd k for which k+2^m is composite for all m < k.

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A206001 Least k such that k*2^m + 1 has a covering set with precisely n primes.

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A244566 Odd integers n such that for every integer k>0, n*2^k+1 has a divisor in the set { 3, 5, 7, 13, 17, 97, 257 }.

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A263169 Smallest positive k such that 78557^k + 2^n is prime, or 0 if no such value exists.

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A263391 Sierpiński numbers that have at least two covering sets.

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A270271 Odd numbers n such that for every k >= 1, n*2^k + 1 has a divisor in the set {3, 5, 17, 257, 641, 65537, 6700417}.

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A270895 Sierpiński numbers that are squares.

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A270896 Values of n for which n^2 is a Sierpiński number.

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A270994 a(n) = 9454129 + 11184810*n.

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