Displaying 71-80 of 82 results found.
Primes p such that (4^p - 2^p + 1)/3 is prime.
+10
1
3, 5, 7, 13, 29, 61, 383, 401, 1637, 1871, 36229, 44771, 44797, 75167
COMMENTS
Terms > 1871 correspond to probable primes.
Is 9 the only composite k such that (4^k - 2^k + 1)/3 is prime? Checked up to 20000. - Andrew Howroyd, Sep 10 2024
EXAMPLE
3 is a term because 3 is prime and (4^3 - 2^3 + 1)/3 = 19 is also prime.
PROG
(PARI) isok(k)={k%2 && ispseudoprime((4^k - 2^k + 1)/3)}
{ forprime(p=3, 2000, if(isok(p), print1(p, ", "))) } \\ Andrew Howroyd, Dec 31 2022
Wagstaff numbers that are of the form 4*k + 3.
+10
1
3, 7, 11, 19, 23, 31, 43, 79, 127, 167, 191, 199, 347, 3539, 5807, 10691, 11279, 12391, 14479, 83339, 117239, 127031, 141079, 269987, 986191, 4031399
COMMENTS
13347311 and 13372531 are also in the sequence, but may not be the next terms.
PROG
(Python)
from itertools import count, islice
from sympy import prime, isprime
def A361562_gen(): # generator of terms return filter(lambda p: p&2 and isprime(((1<<p)+1)//3), (prime(n) for n in count(2)))
Wagstaff numbers that are of the form 4*k + 1.
+10
1
5, 13, 17, 61, 101, 313, 701, 1709, 2617, 10501, 42737, 95369, 138937, 267017, 374321
COMMENTS
15135397 is also in the sequence, but may not be the next term.
PROG
(Python)
from itertools import count, islice
from sympy import prime, isprime
def A361563_gen(): # generator of terms return filter(lambda p: not p&2 and isprime(((1<<p)+1)//3), (prime(n) for n in count(2)))
Numbers n such that 2^n + 2 is an admirable number ( A111592).
+10
0
6, 8, 12, 14, 18, 20, 24, 32, 44, 62, 80, 102, 128, 168, 192, 200, 314, 348, 702
EXAMPLE
a(3)=12 because 2^12 + 2 = 4098 and 1+2+3+683+1366+2049-6 = 4098.
MATHEMATICA
fQ[n_] := Block[{d = Most[ Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Do[ If[ fQ[2^n + 2], Print[n]], {n, 200}] (* Robert G. Wilson v, Aug 30 2005 *)
Nonprime numbers of the form 1 + Sum_{k=1..m} 2^(2*k - 1).
+10
0
171, 10923, 699051, 11184811, 44739243, 178956971, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 11728124029611, 46912496118443, 187649984473771, 750599937895083, 3002399751580331, 12009599006321323
COMMENTS
Prime numbers of the form 1 + Sum_{k=1..m} 2^(2*n - 1) is A000979. Numbers x such that 1 + Sum_{k=1..m} 2^(2*n - 1) is prime for n=1,2,...,x is A127936. A127955 is probably a subset of the present sequence.
MATHEMATICA
a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c] == False, AppendTo[a, c]], {x, 1, 50}]; a
Select[Table[Sum[2^(2k-1), {k, n}]+1, {n, 50}], !PrimeQ[#]&] (* Harvey P. Dale, Dec 23 2017 *)
Least positive number k such that k*p+1 divides 2^p+1 where p is prime(n), or 0 if no such number exists.
+10
0
2, 0, 2, 6, 62, 210, 2570, 9198, 121574, 2, 23091222, 48, 2, 68186767614, 6, 2, 48, 12600235023025650, 109368, 794502, 24, 2550476412689091085878, 6, 2, 10, 8367330694575771627040945250, 4030501264, 6, 955272, 2, 446564785985483547852197647548252246, 8, 8, 32424, 8
COMMENTS
Akin to A186283 except for 2^p+1 and restricted to primes. The larger terms of this sequence occur for the primes p > 3 in sequence A000978. These large terms are (2^p-2)/(3p).
EXAMPLE
2^3+1 = 9 has no factor of the form k*3+1 except 1, so a(primepi(3)) = a(2) = 0.
2^29+1 = 536870913 has factor 2*29+1=59, so a(primepi(29)) = a(10) = 2.
MAPLE
f:= proc(n) local p, F;
p:= ithprime(n);
F:= select(t -> t mod p = 1, numtheory:-divisors(2^p+1) minus {1});
if F = {} then 0 else (min(F)-1)/p; fi
end proc:
MATHEMATICA
Table[q = First /@ FactorInteger[2^p + 1]; s = Select[q, Mod[#1, p] == 1 &, 1]; If[s == {}, 0, (s[[1]] - 1)/p], {p, Prime[Range[30]]}]
Numbers n such that 2^n + 1 can be written in the form a^2 + 5*b^2.
+10
0
2, 3, 7, 10, 11, 19, 23, 31, 43, 47, 50, 58, 71, 79, 82, 107, 127, 167, 178, 179, 191, 199, 250, 290, 298, 311, 347, 359, 410, 487, 563, 599, 683, 751, 802, 890, 907, 1051
COMMENTS
These 2^n + 1 numbers can only have prime factors of the form 1 (mod 20) or 3 (mod 20) or 5 (mod 20) or 7 (mod 20) or 9 (mod 20) raised to an odd power, but their overall product 2^n+1 can only be 1 (mod 20) or 5 (mod 20) or 9 (mod 20). This statement is limited to odd numbers.
In general,
A number n can be written in the form a^2+5*b^2 if and only if n is 0,
or of the form 2^(2i) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m)
or of the form 2^(2i+1) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m+1),
for integers i,j,k,m, for primes p,q.
EXAMPLE
3 is in the sequence because 2^3 + 1 = 9 can be written as 2^2 + 5 * 1^2 = 9.
PROG
(PARI) for(i=2, 500, a=factorint(2^i+1)~; has=0; for(j=1, #a, if(((a[1, j]%20>10)||(i%4<2))&&a[2, j]%2==1, has=1; break)); if(has==0, print(i", ")))
(PARI) for(i=2, 500, a=factorint(2^i+1)~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", ")))
EXTENSIONS
Terms corrected by V. Raman, Sep 20 2012
Numbers n such that (2^n+1)/3 is prime and can be written in the form a^2 + 3*b^2.
+10
0
7, 13, 19, 31, 43, 61, 79, 127, 199, 313, 2617, 10501, 12391, 14479, 138937, 141079, 986191
COMMENTS
The exponent is congruent to 1 mod 3.
Numbers n such that (2^n+1)/3 is prime, but cannot be written in the form a^2 + 3*b^2.
+10
0
5, 11, 17, 23, 101, 167, 191, 347, 701, 1709, 3539, 5807, 10691, 11279, 42737, 83339, 95369, 117239, 127031, 267017, 269987, 374321, 4031399
COMMENTS
The exponent is congruent to 2 mod 3.
Indices of Wagstaff primes.
+10
0
2, 5, 14, 124, 399, 4552, 15898, 203095, 37029521, 105973558438, 19140185454656173, 3827634977577891833517
LINKS
Chris K. Caldwell, Wagstaff, The Top Twenty, The PrimePages.
EXAMPLE
For n = 3 the third Wagstaff prime is A000979(3) = 43 and 43 is also the 14th prime number, so a(3) = 14.
PROG
(PARI) default(primelimit, 10^9); forprime(p=3, 31, q=(2^p+1)/3; if(isprime(q), print1(primepi(q)", "))) \\ Jens Kruse Andersen, Jun 22 2014
EXTENSIONS
a(12) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 05 2024
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