A050526 -id:A050526 - cp's OEIS frontend

A002254 Numbers k such that 5*2^k + 1 is prime.

1, 3, 7, 13, 15, 25, 39, 55, 75, 85, 127, 1947, 3313, 4687, 5947, 13165, 23473, 26607, 125413, 209787, 240937, 819739, 1282755, 1320487, 1777515

Offset: 1

N. J. A. Sloane

H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
C. K. Caldwell, The Prime Pages
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681.
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681. [Annotated scanned copy of selected pages. The first page is (accidentally) included with the scan of the Wilson letter below.]
Eric Weisstein's World of Mathematics, Proth Prime
R. G. Wilson v, Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A050526.

lst={};Do[If[PrimeQ[5*2^n+1], AppendTo[lst, n]], {n, 0, 2*10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)

is(n)=isprime(5*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017

Corrected (removed incorrect term 40937) and added more terms (from http://web.archive.org/web/20161028080239/http://www.prothsearch.net/riesel.html), Joerg Arndt, Apr 07 2013

A083575 a(0) = 6; for n>0, a(n) = 2*a(n-1) - 1.

Original entry on oeis.org

6, 11, 21, 41, 81, 161, 321, 641, 1281, 2561, 5121, 10241, 20481, 40961, 81921, 163841, 327681, 655361, 1310721, 2621441, 5242881, 10485761, 20971521, 41943041, 83886081, 167772161, 335544321, 671088641, 1342177281, 2684354561, 5368709121, 10737418241

Offset: 0

N. J. A. Sloane, Jun 15 2003

The primes in this sequence are listed in A050526. - M. F. Hasler, Oct 30 2010

An Engel expansion of 2/5 to the base 2 as defined in A181565, with the associated series expansion 2/5 = 2/6 + 2^2/(6*11) + 2^3/(6*11*21) + 2^4/(6*11*21*41) + ... . - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A020737, A050526, A083683, A083686, A083705, A168596, A181565, A195744.

[5*2^n+1 : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011

NestList[2#-1&,6,40] (* Harvey P. Dale, Jun 23 2017 *)

PARI
```
a(n)=5<M. F. Hasler, Oct 30 2010
```

Vec((6-7*x)/((1-x)*(1-2*x)) + O(x^40)) \\ Colin Barker, Sep 20 2016

a(n) = 5*2^n + 1. - M. F. Hasler, Oct 30 2010
a(n) = 3*a(n-1) - 2*a(n-2), n>1. - Vincenzo Librandi, Nov 03 2011
G.f.: (6-7*x) / ((1-x)*(1-2*x)). - Colin Barker, Sep 20 2016
E.g.f.: exp(x)*(1 + 5*exp(x)). - Stefano Spezia, Oct 08 2022
Product_{n>=0} (1 + 1/a(n)) = 7/5. - Amiram Eldar, Aug 04 2024

A226366 Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

7, 25, 39, 75, 127, 1947, 3313, 23473, 125413

Offset: 1

Arkadiusz Wesolowski, Jun 05 2013

No other terms below 5330000.

The reason all terms are odd is that if k is even, then 5*2^k + 1 == (-1)*(-1)^k + 1 = (-1)*1 + 1 = 0 (mod 3). So if k is even, then 3 divides 5*2^k + 1, and since 3 divides no other Fermat number than F_0=3 itself, we do not have a Fermat factor. - Jeppe Stig Nielsen, Jul 21 2019

Wilfrid Keller, Fermat factoring status
J. C. Morehead, Note on the factors of Fermat's numbers, Bull. Amer. Math. Soc., Volume 12, Number 9 (1906), pp. 449-451.
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002254.

Cf. A000215, A050526, A057775, A057778, A201364, A204620.

lst = {}; Do[p = 5*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 3313, 2}]; lst

isok(n) = my(p = 5*2^n + 1, z = znorder(Mod(2, p))); isprime(p) && ((z >> valuation(z, 2)) == 1); \\ Michel Marcus, Nov 10 2018

A334092 Primes p of the form of the form q*2^h + 1, where q is one of the Fermat primes; Primes p for which A329697(p) == 2.

Original entry on oeis.org

7, 11, 13, 41, 97, 137, 193, 641, 769, 12289, 40961, 163841, 557057, 786433, 167772161, 2281701377, 3221225473, 206158430209, 2748779069441, 6597069766657, 38280596832649217, 180143985094819841, 221360928884514619393, 188894659314785808547841, 193428131138340667952988161

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p such that p-1 is not a power of two, but for which A171462(p-1) = (p-1-A052126(p-1)) is [a power of 2].

Primes of the form ((2^(2^k))+1)*2^h + 1, where ((2^(2^k))+1) is one of the Fermat primes, A019434, 3, 5, 17, 257, ..., .

Charles R Greathouse IV, Table of n, a(n) for n = 1..53

Primes in A334102.
Intersection of A081091 and A147545.
Subsequences: A039687, A050526, A300407.
Cf. A000040, A010051, A019434, A052126, A171462, A329697. Cf. also A334093, A334094, A334095, A334096.

isA334092(n) = (isprime(n)&&2==A329697(n));

A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
A209229(n) = (n && !bitand(n,n-1));
isA334092(n) = (isprime(n)&&(!A209229(n-1))&&A209229(n-1-A052126(n-1)));

list(lim)=if(exponent(lim\=1)>=2^33, error("Verify composite character of more Fermat primes before checking this high")); my(v=List(),t); for(e=0,4, t=2^2^e+1; while((t<<=1)Charles R Greathouse IV, Apr 14 2020

More terms from Giovanni Resta, Apr 14 2020

A147545 Primes of the form p*2^k+1 with k>0 and p=1 or p in this sequence.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 167, 179, 193, 227, 233, 257, 353, 359, 389, 449, 467, 641, 719, 769, 773, 857, 929, 1097, 1283, 1409, 1433, 1439, 1553, 1697, 1889, 2657, 2819, 2879, 3089, 3329, 3593, 3617, 3779, 5639

Offset: 1

T. D. Noe, Nov 07 2008

nonn

This sequence starts like A074781 but grows much faster. Observe that there can be large differences between consecutive terms. Can it be shown that there is always such a prime between consecutive powers of 2? Or that this sequence is infinite? By theorem 1 of the Noe paper, this sequence is a subsequence of A135832, primes in Section I of the phi iteration.
From Antti Karttunen, Apr 19 2020: (Start)
Sequence can be considered as a generalization of Fermat primes, A019434, which is a subsequence of this sequence.
All terms with binary weight k (A000120, at least 2 for these terms) can be found as a subset of primes found on the row k-1 of array A334100. E.g. primes with weight 2 are Fermat primes (A019434), those with weight 3 are A334092 (which doesn't contain any other primes), those with weight 4 are in A334093 (among also other kind of primes), those with weights 5, 6, 7 are included as (proper) subsets in A334094, A334095 and A334096 respectively. (End)

T. D. Noe, Table of n, a(n) for n=1..2000
T. D. Noe, Primes in classes of the iterated totient function, J. Integer Sequences, 11 (2008), Article 08.1.2.

Cf. A000265, A329697, A334100.
Subsequence of A074781, and of A135832.
Subsequences: A019434, A334092 (including A039687, A050526, A300407).

nn=2^13; t={1}; i=1; While[q=t[[i]]; k=1; While[p=1+q*2^k; p

A000265(n) = (n>>valuation(n,2));
isA147454(n) = ((n>2)&&isprime(n)&&((1==(n=A000265(n-1)))||isA147454(n))); \\ Antti Karttunen, Apr 19 2020

A329697(a(n)) = A000120(a(n)) - 1. - Antti Karttunen, Apr 19 2020

A300407 Primes of the form 17*2^n + 1.

Original entry on oeis.org

137, 557057, 2281701377, 38280596832649217, 3032901347000164747248857685080177164813336577, 240291200809860268823328460101036918152537809975084178304538443375796289537, 4031417378886400659867047414062478199819447786118941877597755244819503521544011777

Offset: 1

Martin Renner, Mar 05 2018

nonn

For the corresponding exponents n see A002259.

			From _Muniru A Asiru_, Mar 29 2018: (Start)
137 is a member because 17 * 2^3 + 1 = 137 which is a prime.
557057 is a member because 17 * 2^15 + 1 = 557057 which is a prime.
2281701377 is a member because 17 * 2^27 + 1 = 2281701377 which is a prime.
... (End)

Ray Ballinger, Wilfried Keller, List of primes k*2^n + 1 for k < 300.

Cf. A002259, A019434, A039687, A050527, A050528, A050529, A195745, A291049, A050526, A300406, A300408.

Filtered(List([1..270],n->17*2^n + 1),IsPrime); # Muniru A Asiru, Mar 06 2018

[a: n in [1..300] | IsPrime(a) where a is 17*2^n + 1]; // Vincenzo Librandi, Mar 07 2018

a:=(n,k)->`if`(isprime(k*2^n+1), k*2^n+1, NULL):
seq(a(n,17), n=1..267);

Select[Table[17 2^n + 1, {n, 400}], PrimeQ] (* Vincenzo Librandi, Mar 07 2018 *)

lista(nn) = {for(k=1, nn, if(ispseudoprime(p=17*2^k+1), print1(p, ", ")));} \\ Altug Alkan, Mar 28 2018

A300406 Primes of the form 13*2^n + 1.

Original entry on oeis.org

53, 3329, 13313, 13631489, 3489660929, 62864142619960717084721153, 5100145160001678120616578906356228963083163798627028041729, 6779255729241169695101387251026410519979286814120235842117075415451380965612384558178346467329, 1735489466685739441945955136262761093114697424414780375581971306355553527196770446893656695635969

Offset: 1

Martin Renner, Mar 05 2018

nonn

For the corresponding exponents n see A032356.

Ray Ballinger, Wilfried Keller, List of primes k*2^n + 1 for k < 300.

Cf. A019434, A039687, A032356, A050527, A050528, A050529, A173236, A195745, A050526, A300407, A300408.

Filtered(List([1..500],n->13*2^n + 1),IsPrime); # Muniru A Asiru, Mar 06 2018

[a: n in [1..400] | IsPrime(a) where a is 13*2^n + 1]; // Vincenzo Librandi, Mar 06 2018

a:=(n,k)->`if`(isprime(k*2^n+1), k*2^n+1, NULL):
seq(a(n,13), n=1..316);

Select[Table[13 2^n + 1, {n, 400}], PrimeQ] (* Vincenzo Librandi, Mar 06 2018 *)

lista(nn) = {for(k=1, nn, if(ispseudoprime(p=13*2^k+1), print1(p, ", ")));} \\ Altug Alkan, Mar 29 2018

a(n) = A168596(A032356(n)). - Michel Marcus, Mar 29 2018

A300408 Primes of the form 19*2^n + 1.

Original entry on oeis.org

1217, 19457, 1337006139375617

Offset: 1

Martin Renner, Mar 05 2018

Next term a(4) = 19*2^366 + 1 > 10^111.

For the corresponding exponents n see A032359.

Ray Ballinger, Wilfried Keller, List of primes k*2^n + 1 for k < 300.

Cf. A019434, A032359, A039687, A050527, A050528, A050529, A195745, A050526, A300406, A300407.

Filtered(List([1..500],n->19*2^n + 1),IsPrime); # Muniru A Asiru, Mar 06 2018

a:=(n,k)->`if`(isprime(k*2^n+1), k*2^n+1, NULL):
seq(a(n,19), n=1..366);

lista(nn) = {for(k=1, nn, if(ispseudoprime(p=19*2^k+1), print1(p, ", ")));} \\ Altug Alkan, Mar 29 2018

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A002254 Numbers k such that 5*2^k + 1 is prime.

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A083575 a(0) = 6; for n>0, a(n) = 2*a(n-1) - 1.

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A226366 Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

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A334092 Primes p of the form of the form q*2^h + 1, where q is one of the Fermat primes; Primes p for which A329697(p) == 2.

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A147545 Primes of the form p*2^k+1 with k>0 and p=1 or p in this sequence.

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A300407 Primes of the form 17*2^n + 1.

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A300406 Primes of the form 13*2^n + 1.

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