A300407 -id:A300407 - cp's OEIS frontend

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.

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

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

Original entry on oeis.org

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

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

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

cp's OEIS Frontend

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

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

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GAP

Magma

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Formula

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

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GAP

Maple

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