Displaying 1-10 of 46 results found.
The number of divisors of n whose prime factors are all of the form 2^k + 1 ( A092506).
+20
1
1, 2, 2, 3, 2, 4, 1, 4, 3, 4, 1, 6, 1, 2, 4, 5, 2, 6, 1, 6, 2, 2, 1, 8, 3, 2, 4, 3, 1, 8, 1, 6, 2, 4, 2, 9, 1, 2, 2, 8, 1, 4, 1, 3, 6, 2, 1, 10, 1, 6, 4, 3, 1, 8, 2, 4, 2, 2, 1, 12, 1, 2, 3, 7, 2, 4, 1, 6, 2, 4, 1, 12, 1, 2, 6, 3, 1, 4, 1, 10, 5, 2, 1, 6, 4, 2, 2, 4, 1, 12, 1, 3, 2, 2, 2, 12, 1, 2, 3, 9
FORMULA
Multiplicative with a(p^e) = e+1 if p is in A092506 (i.e., p is either 2 or a Fermat prime), and 1 otherwise. a(n) >= 1, with equality if and only if all the prime factors of n are not of the form 2^k + 1.
a(n) <= A000005(n), with equality if and only if all the prime factors of n are in A092506 (n is in A143513 assuming that there are only 5 Fermat primes). Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/Product_{k>=1} (1 - 1/ A092506(k)) = 3.99999999906867742538... . This value is exactly 4294967295/1073741824 if there are only 5 Fermat primes.
MATHEMATICA
q[n_] := AllTrue[FactorInteger[n][[;; , 1]], # - 1 == 2^IntegerExponent[# - 1, 2] &]; f[p_, e_] := If[q[p], e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if((f[i, 1]-1) >> valuation(f[i, 1]-1, 2) == 1 , f[i, 2] + 1, 1))};
Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.
+10
157
1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1
COMMENTS
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section). The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270. The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n. Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels. Odd-indexed rows are also the rows of the irregular triangle A237270. Even-indexed rows are also the rows of the triangle A237593. Lengths of the odd-indexed rows are in A237271. Lengths of the even-indexed rows give 2* A003056. Row sums of the odd-indexed rows gives A000203, the sum of divisors function. Row sums of the even-indexed rows give the positive even numbers (see A005843). From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
EXAMPLE
Irregular triangle begins:
1;
1, 1;
3;
2, 2;
2, 2;
2, 1, 1, 2;
7;
3, 1, 1, 3;
3, 3;
3, 2, 2, 3;
12;
4, 1, 1, 1, 1, 4;
4, 4;
4, 2, 1, 1, 2, 4;
15;
5, 2, 1, 1, 2, 5;
5, 3, 5;
5, 2, 2, 2, 2, 5;
9, 9;
6, 2, 1, 1, 1, 1, 2, 6;
6, 6;
6, 3, 1, 1, 1, 1, 3, 6;
28;
7, 2, 2, 1, 1, 2, 2, 7;
7, 7;
7, 3, 2, 1, 1, 2, 3, 7;
12, 12;
8, 3, 1, 2, 2, 1, 3, 8;
8, 8, 8;
8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
31;
9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
1 1 = 1 |_| | | | | | | | | | | | | | | |
2 3 = 3 |_ _|_| | | | | | | | | | | | | |
3 4 = 2 + 2 |_ _| _|_| | | | | | | | | | | |
4 7 = 7 |_ _ _| _|_| | | | | | | | | |
5 6 = 3 + 3 |_ _ _| _| _ _|_| | | | | | | |
6 12 = 12 |_ _ _ _| _| | _ _|_| | | | | |
7 8 = 4 + 4 |_ _ _ _| |_ _|_| _ _|_| | | |
8 15 = 15 |_ _ _ _ _| _| | _ _ _|_| |
9 13 = 5 + 3 + 5 |_ _ _ _ _| | _|_| | _ _ _|
10 18 = 9 + 9 |_ _ _ _ _ _| _ _| _| |
11 12 = 6 + 6 |_ _ _ _ _ _| | _| _| _|
12 28 = 28 |_ _ _ _ _ _ _| |_ _| _|
13 14 = 7 + 7 |_ _ _ _ _ _ _| | _ _|
14 24 = 12 + 12 |_ _ _ _ _ _ _ _| |
15 24 = 8 + 8 + 8 |_ _ _ _ _ _ _ _| |
16 31 = 31 |_ _ _ _ _ _ _ _ _|
...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
Level _ _
1 _|1|1|_
2 _|2 _|_ 2|_
3 _|2 |1|1| 2|_
4 _|3 _|1|1|_ 3|_
5 _|3 |2 _|_ 2| 3|_
6 _|4 _|1|1|1|1|_ 4|_
7 _|4 |2 |1|1| 2| 4|_
8 _|5 _|2 _|1|1|_ 2|_ 5|_
9 _|5 |2 |2 _|_ 2| 2| 5|_
10 _|6 _|2 |1|1|1|1| 2|_ 6|_
11 _|6 |3 _|1|1|1|1|_ 3| 6|_
12 _|7 _|2 |2 |1|1| 2| 2|_ 7|_
13 _|7 |3 |2 _|1|1|_ 2| 3| 7|_
14 _|8 _|3 _|1|2 _|_ 2|1|_ 3|_ 8|_
15 _|8 |3 |2 |1|1|1|1| 2| 3| 8|_
16 |9 |3 |2 |1|1|1|1| 2| 3| 9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n. The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n). The total number of vertical line segments in the n-th level of the diagram equals A131507(n). The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - Omar E. Pol, Aug 28 2018 The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - Omar E. Pol, Nov 09 2022
CROSSREFS
Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871. Cf. A000079 (powers of 2)........., for the characteristic shape see A346872. Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level. Cf. A000217 (triangular numbers).., for the characteristic shape see A346873. Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875. Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876. Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level. Cf. A008586 (multiples of 4)......, perimeters of the successive levels. Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613. Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo. Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864. Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level. Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others). Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238. Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.
Numbers of edges of regular polygons constructible with ruler (or, more precisely, an unmarked straightedge) and compass.
(Formerly M0505)
+10
42
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285
COMMENTS
The terms 1 and 2 correspond to degenerate polygons.
The sequence can be also defined as follows: (i) 1 is a member. (ii) Double of any member is also a member. (iii) If a member is not divisible by a Fermat prime F_k then its product with F_k is also a member. In particular, the powers of 2 ( A000079) are a subset and so are the Fermat primes ( A019434), which are the only odd prime members. The definition is too restrictive (though correct): The Georg Mohr - Lorenzo Mascheroni theorem shows that constructibility using a straightedge and a compass is equivalent to using compass only. Moreover, Jean Victor Poncelet has shown that it is also equivalent to using straightedge and a fixed ('rusty') compass. With the work of Jakob Steiner, this became part of the Poncelet-Steiner theorem establishing the equivalence to using straightedge and a fixed circle (with a known center). A further extension by Francesco Severi replaced the availability of a circle with that of a fixed arc, no matter how small (but still with a known center).
Constructibility implies that when m is a member of this sequence, the edge length 2*sin(Pi/m) of an m-gon with circumradius 1 can be written as a finite expression involving only integer numbers, the four basic arithmetic operations, and the square root. (End)
If x,y are terms, and gcd(x,y) is a power of 2 then x*y is also a term. - David James Sycamore, Aug 24 2024
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 183.
Allan Clark, Elements of Abstract Algebra, Chapter 4, Galois Theory, Dover Publications, NY 1984, page 124.
Duane W. DeTemple, "Carlyle circles and the Lemoine simplicity of polygon constructions." The American Mathematical Monthly 98.2 (1991): 97-108. - N. J. A. Sloane, Aug 05 2021 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. L. van der Waerden, Modern Algebra. Unger, NY, 2nd ed., Vols. 1-2, 1953, Vol. 1, p. 187.
LINKS
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 463. Original (in Latin).
FORMULA
Terms from 3 onward are computable as numbers such that cototient-of-totient equals the totient-of-totient: Flatten[Position[Table[co[eu[n]]-eu[eu[n]], {n, 1, 10000}], 0]] eu[m]=EulerPhi[m], co[m]=m-eu[m]. - Labos Elemer, Oct 19 2001, clarified by Antti Karttunen, Nov 27 2017 If the well-known conjecture that there are only five prime Fermat numbers F_k=2^{2^k}+1, k=0,1,2,3,4 is true, then we have exactly: Sum_{n>=1} 1/a(n)= 2*Product_{k=0..4} (1+1/F_k) = 4869735552/1431655765 = 3.40147098978.... - Vladimir Shevelev and T. D. Noe, Dec 01 2010 log a(n) >> sqrt(n); if there are finitely many Fermat primes, then log a(n) ~ k log n for some k. - Charles R Greathouse IV, Oct 23 2015
EXAMPLE
34 is a term of this sequence because a circle can be divided into exactly 34 parts. 7 is not.
MATHEMATICA
Select[ Range[ 1300 ], IntegerQ[ Log[ 2, EulerPhi[ # ] ] ]& ] (* Olivier Gérard Feb 15 1999 *) (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Take[ Union[ Flatten[ NestList[2# &, Times @@@ Table[ UnrankSubset[n, Join[{1}, Table[2^2^i + 1, {i, 0, 4}]]], {n, 63}], 11]]], 60] (* Robert G. Wilson v, Jun 11 2005 *) nn=10; logs=Log[2, {2, 3, 5, 17, 257, 65537}]; lim2=Floor[nn/logs[[1]]]; Sort[Reap[Do[z={i, j, k, l, m, n}.logs; If[z<=nn, Sow[2^z]], {i, 0, lim2}, {j, 0, 1}, {k, 0, 1}, {l, 0, 1}, {m, 0, 1}, {n, 0, 1}]][[2, 1]]]
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 1300; Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}] (* Robert G. Wilson v, Jul 28 2014 *)
PROG
(Haskell)
a003401 n = a003401_list !! (n-1)
a003401_list = map (+ 1) $ elemIndices 1 $ map a209229 a000010_list
(PARI) for(n=1, 10^4, my(t=eulerphi(n)); if(t/2^valuation(t, 2)==1, print1(n, ", "))); \\ Joerg Arndt, Jul 29 2014 (PARI) is(n)=n>>=valuation(n, 2); if(n<7, return(n>0)); my(k=logint(logint(n, 2), 2)); if(k>32, my(p=2^2^k+1); if(n%p, return(0)); n/=p; unknown=1; if(n%p==0, return(0)); p=0; if(is(n)==0, 0, "unknown [has large Fermat number in factorization]"), 4294967295%n==0) \\ Charles R Greathouse IV, Jan 09 2022 (PARI) is(n)=n>>=valuation(n, 2); 4294967295%n==0 \\ valid for n <= 2^2^33, conjecturally valid for all n; Charles R Greathouse IV, Jan 09 2022 (Python)
from sympy import totient
A003401_list = [n for n in range(1, 10**4) if format(totient(n), 'b').count('1') == 1]
CROSSREFS
Positions of zeros in A293516 (apart from two initial -1's), and in A336469, positions of ones in A295660 and in A336477 (characteristic function).
Powers of 2 and odd primes; alternatively, numbers that cannot be written as a sum of at least three consecutive positive integers.
+10
29
1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 256
COMMENTS
Also the odd noncomposite numbers ( A006005) and the powers of 2 with positive exponent, in increasing order. If a(n) is composite and a(n) - a(n-1) = 1 then a(n-1) is a Mersenne prime ( A000668), hence a(n-1)*a(n)/2 is a perfect number ( A000396) and a(n-1)*a(n) equals the sum of divisors of a(n-1)*a(n)/2. If a(n) is even and a(n+1) - a(n) = 1 then a(n+1) is a Fermat prime ( A019434). (End)
MAPLE
N:= 300: # to get all terms <= N
S:= {seq(2^i, i=0..ilog2(N))} union select(isprime, { 2*i+1 $ i=1..floor((N-1)/2) }):
MATHEMATICA
a[n_] := Product[GCD[2 i - 1, n], {i, 1, (n - 1)/2}] - 1;
PROG
(Python)
from sympy import primepi
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n+x+(0 if x<=1 else 1-primepi(x))-x.bit_length())
(PARI) list(lim)=Set(concat(concat(1, primes(lim)), vector(logint(lim\2, 2), i, 2^(i+1)))) \\ Charles R Greathouse IV, Sep 19 2024
EXTENSIONS
This entry is the result of merging an old incorrect entry and a more recent correct version. N. J. A. Sloane, Dec 07 2015
Primes of form 1+(2^a)*(3^b), a>0, b>0.
+10
20
7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993
COMMENTS
Prime numbers n such that cos(2*Pi/n) is an algebraic number of a 3-smooth degree, but not a 2-smooth degree. - Artur Jasinski, Dec 13 2006 In this case gcd(a,b) is a power of 2.
A regular polygon of n sides is constructible by paper folding if and only if n=2^r3^sp_1...p_t with p_i being distinct primes of this kind. (End)
MAPLE
N:= 10^10: # to get all terms <= N+1
sort(select(isprime, [seq(seq(1+2^a*3^b, a=1..ilog2(N/3^b)), b=1..floor(log[3](N)))])); # Robert Israel, Mar 02 2018
MATHEMATICA
Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]], 1] == {3} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *) mx = 1500000; s = Sort@ Flatten@ Table[1 + 2^j*3^k, {j, Log[2, mx]}, {k, Log[3, mx/2^j]}]; Select[s, PrimeQ] (* Robert G. Wilson v, Sep 28 2012 *) Select[Prime[Range[114000]], FactorInteger[#-1][[All, 1]]=={2, 3}&] (* Harvey P. Dale, Apr 13 2019 *)
Prime numbers of the form 4^k + 1.
+10
8
COMMENTS
Similar to Fermat primes, A019434, a(n) is of the form (2^2)^k + 1.
MATHEMATICA
Select[4^Range[0, 100] + 1, PrimeQ]
Primes of the form 4^k + 3.
+10
5
7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, 19342813113834066795298819
EXAMPLE
67 is a term because 4^3 + 3 = 67 is prime.
MATHEMATICA
Select[Table[4^n + 3, {n, 0, 200}], PrimeQ]
PROG
(Magma) [a: n in [0..200] | IsPrime(a) where a is 4^n+3];
CROSSREFS
Cf. Primes of the form r^k + h: A092506 (r=2, h=1), A057733 (r=2, h=3), A123250 (r=2, h=5), A104066 (r=2, h=7), A104070 (r=2, h=9), A057735 (r=3, h=2), A102903 (r=3, h=4), A102870 (r=3, h=8), A102907 (r=3, h=10), A290200 (r=4, h=1), this sequence (r=4, h=3), A228027 (r=4, h=9), A182330 (r=5, h=2), A228029 (r=5, h=6), A102910 (r=5, h=8), A182331 (r=6, h=1), A104118 (r=6, h=5), A104115 (r=6, h=7), A104065 (r=7, h=4), A228030 (r=7, h=6), A228031 (r=7, h=10), A228032 (r=8, h=3), A228033 (r=8, h=5), A144360 (r=8, h=7), A145440 (r=8, h=9), A228034 (r=9, h=2), A159352 (r=10, h=3), A159031 (r=10, h=7).
Primes of the form 8^n + 3.
+10
5
11, 67, 4099, 32771, 262147, 1073741827, 19342813113834066795298819
MATHEMATICA
Select[Table[8^n + 3, {n, 0, 300}], PrimeQ]
PROG
(Magma) [a: n in [0..300] | IsPrime(a) where a is 8^n+3];
CROSSREFS
Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=8, h=3), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).
Primes of the form 2^n + 39.
+10
5
41, 43, 47, 71, 103, 167, 1063, 2087, 8231, 131111, 536870951, 8589934631, 549755813927, 8796093022247, 154742504910672534362390567, 40564819207303340847894502572071, 162259276829213363391578010288167, 2722258935367507707706996859454145691687
COMMENTS
Associated n: 1, 2, 3, 5, 6, 7, 10, 11, 13, 17, 29, 33, 39, 43, 87, 105, 107, 131, 253, 329, ....
MATHEMATICA
Select[Table[2^n + 39, {n, 0, 500}], PrimeQ]
PROG
(Magma) [a: n in [0..500] | IsPrime(a) where a is 2^n+39];
CROSSREFS
Cf. primes of the form 2^n+k: A092506 (k=1), A057733 (k=3), A123250 (k=5), A104066 (k=7), A104070 (k=9), A156940 (k=11), A104067 (k=13), A144487 (k=15), A156973 (k=17), A104068 (k=19), A156983 (k=21), A176922 (k=23), A104072 (k=25), A104071 (k=27), A156974 (k=29), A104069 (k=31), A176926 (k=33), A176927 (k=35), A176924 (k=37), this sequence (k=39), A176925 (k=41), A243430 (k=43), A243431 (k=45), A243432 (k=47), A104073 (k=49).
Numbers k such that exactly one of 2^k - 1 and 2^k + 1 is a prime.
+10
4
0, 1, 3, 4, 5, 7, 8, 13, 16, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917
COMMENTS
Apart from the first term, all terms are primes (Mersenne exponents) or powers of two (Fermat exponents). The sequence consists of all members of A000043 and A092506, apart from 2. - Charles R Greathouse IV, Mar 20 2010 Numbers k such that one of 2^k+1 or 2^k-1 is prime, but not both. - R. J. Mathar, Mar 29 2010 The sequence "Numbers k such that 2^k + (-1)^k is a prime" gives essentially the same sequence, except with the initial 1 replaced by 2. - Thomas Ordowski, Dec 26 2016 The union of 2 and this sequence gives the values k for which 2^k or 2^k - 1 are the numbers in A006549. - Gionata Neri, Dec 19 2015 The union of 2 and this sequence is the values k for which either 2^k - 1 or 2^k + 1, or both, are prime. The reason why this only yields one additional term, 2, is because the number 3 always divides either 2^k - 1 or 2^k + 1 (also implicit in Ordowski comment). - Jeppe Stig Nielsen, Feb 19 2023
EXAMPLE
0 is in the sequence because 2^0 - 1 = 0 is nonprime and 2^0 + 1 = 2 is prime; 2 is not in the sequence because 2^2 - 1 = 3 and 2^2 + 1 = 5 are both prime.
MATHEMATICA
Select[Range[0, 5000], Xor[PrimeQ[2^# - 1], PrimeQ[2^# + 1]] &] (* Michael De Vlieger, Jan 03 2016 *)
PROG
(PARI) isok(k) = my(p = 2^k-1, q = p+2); bitxor(isprime(p), isprime(q)); \\ Michel Marcus, Jan 03 2016
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