A050527 -id:A050527 - cp's OEIS frontend

A050526 Primes of form 5*2^n+1.

11, 41, 641, 40961, 163841, 167772161, 2748779069441, 180143985094819841, 188894659314785808547841, 193428131138340667952988161, 850705917302346158658436518579420528641

Offset: 1

N. J. A. Sloane, Dec 29 1999

nonn

All terms are odd since if n is even, then 5*2^n+1 is divisible by 3. - Michele Fabbrini, Jun 06 2021

Muniru A Asiru, Table of n, a(n) for n = 1..12
Ray Ballinger, Wilfried Keller, List of primes k*2^n + 1 for k < 300
Wikipedia, Proth's theorem
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

For the corresponding exponents n see A002254.

Cf. A019434, A039687, A050527, A050528, A050529, A195745, A300406, A300407, A300408.

Filtered(List([1..270], n->5*2^n + 1), IsPrime); # Muniru A Asiru, Mar 06 2018

[a: n in [1..200] | IsPrime(a) where a is 5*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, 5), n=1..127); # Martin Renner, Mar 05 2018

lista(nn) = {for(k=1, nn, if(ispseudoprime(p=5*2^k+1), print1(p, ", "))); } \\ Altug Alkan, Mar 29 2018

a(n) = A083575(A002254(n)). - Michel Marcus, Mar 29 2018

A032353 Numbers k such that 7*2^k+1 is prime.

Original entry on oeis.org

2, 4, 6, 14, 20, 26, 50, 52, 92, 120, 174, 180, 190, 290, 320, 390, 432, 616, 830, 1804, 2256, 6614, 13496, 15494, 16696, 22386, 54486, 88066, 95330, 207084, 283034, 561816, 804534, 811230, 1491852, 2139912, 2167800, 2915954, 3015762, 3511774, 5775996

Offset: 1

James R. Buddenhagen

Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
Eric Weisstein's World of Mathematics, Proth Prime
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002255, A050527.

[n: n in [1..830] | IsPrime(7*2^n+1)]; // Arkadiusz Wesolowski, Feb 24 2017

Select[Range[10^3], PrimeQ[7*2^# + 1] &] (* Michael De Vlieger, Feb 25 2017 *)

is(n)=ispseudoprime(7*2^n+1) \\ Charles R Greathouse IV, Feb 20 2017

Added more terms (from http://web.archive.org/web/20161028080239/http://www.prothsearch.net/riesel.html), Joerg Arndt, Apr 07 2013

a(41) from Jeppe Stig Nielsen, Jul 25 2019

A037178 Longest cycle when squaring modulo n-th prime.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 1, 6, 10, 3, 4, 6, 4, 6, 11, 12, 28, 4, 10, 12, 6, 12, 20, 10, 2, 20, 8, 52, 18, 3, 6, 12, 8, 22, 36, 20, 12, 54, 82, 14, 11, 12, 36, 2, 21, 30, 12, 36, 28, 18, 28, 24, 4, 100, 1, 130, 66, 36, 22, 12, 46, 9, 24, 20, 12, 39, 20, 6, 172, 28, 10, 178, 60, 10, 18

Offset: 1

Jud McCranie

nonn

a(n)=1 for Fermat primes, A019434. a(n)=2 for primes in A039687. a(n)=3 for primes in A050527. Sequence A141305 gives those primes p > 3 having the longest possible cycle, (p-3)/2. - T. D. Noe, Jun 24 2008

T. D. Noe, Table of n, a(n) for n=1..10000
E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie, On a digraph defined by squaring modulo n, Fibonacci Quart. 30 (Nov. 1992), 322-333.
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016. See table on page 2.

Cf. A002322, A007733, A256608.
Cf. A019434, A039687, A050527, A141305.
Cf. A037179, A037180.
a(n) = maximal entry in row p of A278185.

a[n_] := Module[{p = Prime[n], k}, k = (p-1)/2^IntegerExponent[p-1, 2]; MultiplicativeOrder[2, k]]; Array[a, 100] (* Jean-François Alcover, Jan 28 2016, after T. D. Noe *)

a(n) = {ppn = prime(n) - 1; k = ppn >> valuation(ppn, 2); znorder(Mod(2, k));} \\ Michel Marcus, Nov 11 2015

rpsi(n) = lcm(znstar(n)[2]); \\ A002322
pb(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ A007733
a(n) = pb(rpsi(prime(n))); \\ Michel Marcus, Jan 28 2016

Let p=prime(n) and k=A000265(p-1), the odd part of p-1. Then a(n) = ord(2,k), that is, the smallest positive integer x such that 2^x = 1 (mod k). - T. D. Noe, Jun 24 2008
a(n) = A007733(A002322(prime(n))). - Michel Marcus, Jan 28 2016
a(n) = A256608(prime(n)).

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

A280003 Numbers k such that 7*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

14, 120, 290, 320, 95330, 2167800

Offset: 1

Arkadiusz Wesolowski, Feb 21 2017

18233956 belongs to this sequence, but its position is currently unknown. - Jeppe Stig Nielsen, Oct 05 2020

Proth Search Page, Prime factors of Fermat numbers
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A032353.

Cf. A000215, A050527, A057778, A201364, A204620, A226366, A280004.

IsInteger := func; [n: n in [1..320] | IsPrime(k) and IsInteger(Log(2, Modorder(2, k))) where k is 7*2^n+1];

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

A322916 Numbers k such that 303*2^k+1 is prime.

Original entry on oeis.org

1, 2, 5, 8, 9, 10, 13, 17, 21, 30, 38, 93, 125, 140, 170, 178, 181, 394, 453, 588, 1161, 1221, 1573, 1665, 1745, 3613, 3661, 5750, 6002, 6198, 6393, 6764, 7209, 7514, 9444, 13034, 15277, 16070, 20042, 22970, 22977, 25674, 28438, 31040, 35137, 42410, 60285

Offset: 1

Robert Price, Dec 30 2018

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..76 (terms n = 1..68 from Robert Price)
Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for 300 < k < 600
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
Eric Weisstein's World of Mathematics, Proth Prime
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002255, A050527.

select(n->isprime(303*2^n+1),[$1..1000]); # Muniru A Asiru, Dec 31 2018

Select[Range[1000], PrimeQ[303*2^# + 1] &] (* Robert Price, Dec 30 2018 *)

A322949 Numbers k such that 315*2^k+1 is prime.

Original entry on oeis.org

1, 3, 6, 9, 20, 22, 27, 32, 72, 97, 99, 104, 107, 120, 140, 142, 151, 180, 304, 305, 342, 440, 489, 521, 635, 665, 673, 767, 876, 1040, 1313, 1359, 1764, 2032, 2224, 2280, 2783, 2832, 2875, 5256, 8225, 10297, 11124, 12124, 17552, 18592, 24435, 30704, 37467

Offset: 1

Robert Price, Dec 31 2018

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..77 (terms n = 1..74 from Robert Price)
Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for 300 < k < 600
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
Eric Weisstein's World of Mathematics, Proth Prime
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002255, A050527.

Select[Range[1000], PrimeQ[315*2^# + 1] &] (* Robert Price, Dec 31 2018 *)

A322957 Numbers k such that 329*2^k+1 is prime.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 27, 55, 87, 105, 111, 163, 225, 311, 487, 537, 723, 735, 771, 1515, 1599, 4685, 5523, 5895, 9723, 20107, 55035, 66355, 108393, 181189, 455645, 604999, 623005, 829207, 1019093, 1246017, 2099771, 2163717, 2266631, 2348105, 2688221, 3002295

Offset: 1

Robert Price, Dec 31 2018

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..45
Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for 300 < k < 600
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
Eric Weisstein's World of Mathematics, Proth Prime
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002255, A050527.

select(n->isprime(329*2^n+1),[$1..1000]); # Muniru A Asiru, Dec 31 2018

Select[Range[1000], PrimeQ[329*2^# + 1] &] (* Robert Price, Dec 31 2018 *)

a(37)-a(41) from Jeppe Stig Nielsen, Jan 04 2020

a(42) from Jeppe Stig Nielsen, Dec 20 2024

cp's OEIS Frontend

A050526 Primes of form 5*2^n+1.

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A032353 Numbers k such that 7*2^k+1 is prime.

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A037178 Longest cycle when squaring modulo n-th prime.

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

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A280003 Numbers k such that 7*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

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

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GAP

Magma

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

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