A053576 -id:A053576 - cp's OEIS frontend

A001317 Sierpiński's triangle (Pascal's triangle mod 2) converted to decimal.

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295, 4294967297, 12884901891, 21474836485, 64424509455, 73014444049, 219043332147, 365072220245, 1095216660735, 1103806595329, 3311419785987

Offset: 0

N. J. A. Sloane

The members are all palindromic in binary, i.e., a subset of A006995. - Ralf Stephan, Sep 28 2004
J. H. Conway writes (in Math Forum): at least the first 31 numbers give odd-sided constructible polygons. See also A047999. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 19 2005 [This observation was also made in 1982 by N. L. White (see letter). - N. J. A. Sloane, Jun 15 2015]
Decimal number generated by the binary bits of the n-th generation of the Rule 60 elementary cellular automaton. Thus: 1; 0, 1, 1; 0, 0, 1, 0, 1; 0, 0, 0, 1, 1, 1, 1; 0, 0, 0, 0, 1, 0, 0, 0, 1; ... . - Eric W. Weisstein, Apr 08 2006
Limit_{n->oo} log(a(n))/n = log(2). - Bret Mulvey, May 17 2008
Equals row sums of triangle A166548; e.g., 17 = (2 + 4 + 6 + 4 + 1). - Gary W. Adamson, Oct 16 2009
Equals row sums of triangle A166555. - Gary W. Adamson, Oct 17 2009
For n >= 1, all terms are in A001969. - Vladimir Shevelev, Oct 25 2010
Let n,m >= 0 be such that no carries occur when adding them. Then a(n+m) = a(n)*a(m). - Vladimir Shevelev, Nov 28 2010
Let phi_a(n) be the number of a(k) <= a(n) and respectively prime to a(n) (i.e., totient function over {a(n)}). Then, for n >= 1, phi_a(n) = 2^v(n), where v(n) is the number of 0's in the binary representation of n. - Vladimir Shevelev, Nov 29 2010
Trisection of this sequence gives rows of A008287 mod 2 converted to decimal. See also A177897, A177960. - Vladimir Shevelev, Jan 02 2011
Converting the rows of the powers of the k-nomial (k = 2^e where e >= 1) term-wise to binary and reading the concatenation as binary number gives every (k-1)st term of this sequence. Similarly with powers p^k of any prime. It might be interesting to study how this fails for powers of composites. - Joerg Arndt, Jan 07 2011
This sequence appears in Pascal's triangle mod 2 in another way, too. If we write it as
  1111111...
  10101010...
  11001100...
  10001000...
we get (taking the period part in each row):
  .(1) (base 2) = 1
  .(10) = 2/3
  .(1100) = 12/15 = 4/5
  .(1000) = 8/15
The k-th row, treated as a binary fraction, seems to be equal to 2^k / a(k). - Katarzyna Matylla, Mar 12 2011
From Daniel Forgues, Jun 16-18 2011: (Start)
Since there are 5 known Fermat primes, there are 32 products of distinct Fermat primes (thus there are 31 constructible odd-sided polygons, since a polygon has at least 3 sides). a(0)=1 (empty product) and a(1) to a(31) are those 31 non-products of distinct Fermat primes.
It can be proved by induction that all terms of this sequence are products of distinct Fermat numbers (A000215):
a(0)=1 (empty product) are products of distinct Fermat numbers in { };
a(2^n+k) = a(k) * (2^(2^n)+1) = a(k) * F_n, n >= 0, 0 <= k <= 2^n - 1.
Thus for n >= 1, 0 <= k <= 2^n - 1, and
  a(k) = Product_{i=0..n-1} F_i^(alpha_i), alpha_i in {0, 1},
this implies
  a(2^n+k) = Product_{i=0..n-1} F_i^(alpha_i) * F_n, alpha_i in {0, 1}.
(Cf. OEIS Wiki links below.)  (End)
The bits in the binary expansion of a(n) give the coefficients of the n-th power of polynomial (X+1) in ring GF(2)[X]. E.g., 3 ("11" in binary) stands for (X+1)^1, 5 ("101" in binary) stands for (X+1)^2 = (X^2 + 1), and so on. - Antti Karttunen, Feb 10 2016

			Given a(5)=51, a(6)=85 since a(5) XOR 2*a(5) = 51 XOR 102 = 85.
From _Daniel Forgues_, Jun 18 2011: (Start)
  a(0) = 1 (empty product);
  a(1) = 3 = 1 * F_0 = a(2^0+0) = a(0) * F_0;
  a(2) = 5 = 1 * F_1 = a(2^1+0) = a(0) * F_1;
  a(3) = 15 = 3 * 5 = F_0 * F_1 = a(2^1+1) = a(1) * F_1;
  a(4) = 17 = 1 * F_2 = a(2^2+0) = a(0) * F_2;
  a(5) = 51 = 3 * 17 = F_0 * F_2 = a(2^2+1) = a(1) * F_2;
  a(6) = 85 = 5 * 17 = F_1 * F_2 = a(2^2+2) = a(2) * F_2;
  a(7) = 255 = 3 * 5 * 17 = F_0 * F_1 * F_2 = a(2^2+3) = a(3) * F_2;
  ... (End)

Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 113.
Henry Wadsworth Gould, Exponential Binomial Coefficient Series, Tech. Rep. 4, Math. Dept., West Virginia Univ., Morgantown, WV, Sept. 1961.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.

Amiram Eldar, Table of n, a(n) for n = 0..3321 (terms 0..300 from T. D. Noe, terms 301..1023 from Tilman Piesk)
Gary W. Adamson, Letter to N. J. A. Sloane, Apr 29 1994, and MS "Algorithm for the Chinese Remainder Theorem".
Gary W. Adamson and N. J. A. Sloane, Correspondence, May 1994, including Adamson's MSS "Algorithm for Generating nth Row of Pascal's Triangle, mod 2, from n", and "The Tower of Hanoi Wheel".
Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104), No. 1 (2013), pp. 73-98; alternative link.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, Vol. 63, No. 1 (1990), pp. 3-20. [Annotated scanned copy]
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Denton Hewgill, A relationship between Pascal's triangle and Fermat numbers, Fib. Quart., Vol. 15, No. 2 (1977), pp. 183-184.
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, Fig. 4.
Dr. Math, Regular polygon formulas.
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
OEIS Wiki, Constructible odd-sided polygons and Sierpinski's triangle.
Vladimir Shevelev, On Stephan's conjectures concerning Pascal triangle modulo 2, J. Alg. Number Theory, Vol. 7, No. 1 (2012) pp. 11-29, preprint, arXiv:1011.6083 [math.NT], 2010-2012.
John Frederick Sweeney, Clifford Clock and the Moolakaprithi Cube, 2014. See p. 26. - _N. J. A. Sloane_, Mar 20 2014
Eric Weisstein's World of Mathematics, Rule 60 and Rule 102.
Neil L. White, Letter to N. J. A. Sloane, Apr 15 1982.
R. G. Wilson, V, Letter to N. J. A. Sloane, Aug. 1993.
Index entries for sequences related to cellular automata
Index entries for sequences operating on polynomials in ring GF(2)[X]

Cf. A000079, A000215 (Fermat numbers), A047999, A048720, A054432, A173019, A177882, A177897, A177960, A193231, A230116, A247282, A249184, A268391.
Cf. A038183 (odd bisection, 1D Cellular Automata Rule 90).
Iterates of A048724 (starting from 1).
Row 3 of A048723.
Positions of records in A268389.
Positions of ones in A268669 and A268384 (characteristic function).
Not the same as A045544 nor as A053576.
Cf. A045544.

a001317 = foldr (\u v-> 2*v + u) 0 . map toInteger . a047999_row
-- Reinhard Zumkeller, Nov 24 2012
(Scheme, with memoization-macro definec, two variants)
(definec (A001317 n) (if (zero? n) 1 (A048724 (A001317 (- n 1)))))
(definec (A001317 n) (if (zero? n) 1 (A048720bi 3 (A001317 (- n 1))))) ;; Where A048720bi implements the dyadic function given in A048720.
;; Antti Karttunen, Feb 10 2016

[&+[(Binomial(n, i) mod 2)*2^i: i in [0..n]]: n in [0..41]]; // Vincenzo Librandi, Feb 12 2016

A001317 := proc(n) local k; add((binomial(n,k) mod 2)*2^k, k=0..n); end;

a[n_] := Nest[ BitXor[#, BitShiftLeft[#, 1]] &, 1, n]; Array[a, 42, 0] (* Joel Madigan (dochoncho(AT)gmail.com), Dec 03 2007 *)
NestList[BitXor[#,2#]&,1,50] (* Harvey P. Dale, Aug 02 2021 *)

PARI
```
a(n)=sum(i=0,n,(binomial(n,i)%2)*2^i)
```

a=1; for(n=0, 66, print1(a,", "); a=bitxor(a,a<<1) ); \\ Joerg Arndt, Mar 27 2013

A001317(n,a=1)={for(k=1,n,a=bitxor(a,a<<1));a} \\ M. F. Hasler, Jun 06 2016

a(n) = subst(lift(Mod(1+'x,2)^n), 'x, 2); \\ Gheorghe Coserea, Nov 09 2017

from sympy import binomial
def a(n): return sum([(binomial(n, i)%2)*2**i for i in range(n + 1)]) # Indranil Ghosh, Apr 11 2017

def A001317(n): return int(''.join(str(int(not(~n&k))) for k in range(n+1)),2) # Chai Wah Wu, Feb 04 2022

a(n+1) = a(n) XOR 2*a(n), where XOR is binary exclusive OR operator. - Paul D. Hanna, Apr 27 2003
a(n) = Product_{e(j, n) = 1} (2^(2^j) + 1), where e(j, n) is the j-th least significant digit in the binary representation of n (Roberts: see Allouche & Shallit). - Benoit Cloitre, Jun 08 2004
a(2*n+1) = 3*a(2*n). Proof: Since a(n) = Product_{k in K} (1 + 2^(2^k)), where K is the set of integers such that n = Sum_{k in K} 2^k, clearly K(2*n+1) = K(2*n) union {0}, hence a(2*n+1) = (1+2^(2^0))*a(2*n) = 3*a(2*n). - Emmanuel Ferrand and Ralf Stephan, Sep 28 2004
a(32*n) = 3 ^ (32 * n * log(2) / log(3)) + 1. - Bret Mulvey, May 17 2008
For n >= 1, A000120(a(n)) = 2^A000120(n). - Vladimir Shevelev, Oct 25 2010
a(2^n) = A000215(n); a(2^n-1) = a(2^n)-2; for n >= 1, m >= 0,
a(2^(n-1)-1)*a(2^n*m + 2^(n-1)) = 3*a(2^(n-1))*a(2^n*m + 2^(n-1)-2). - Vladimir Shevelev, Nov 28 2010
Sum_{k>=0} 1/a(k) = Product_{n>=0} (1 + 1/F_n), where F_n=A000215(n);
Sum_{k>=0} (-1)^(m(k))/a(k) = 1/2, where {m(n)} is Thue-Morse sequence (A010060).
If F_n is defined by F_n(z) = z^(2^n) + 1 and a(n) by (1/2)*Sum_{i>=0}(1-(-1)^{binomial(n,i)})*z^i, then, for z > 1, the latter two identities hold as well with the replacement 1/2 in the right hand side of the 2nd one by 1-1/z. - Vladimir Shevelev, Nov 29 2010
G.f.: Product_{k>=0} ( 1 + z^(2^k) + (2*z)^(2^k) ). - conjectured by Shamil Shakirov, proved by Vladimir Shevelev
a(n) = A000225(n+1) - A219843(n). - Reinhard Zumkeller, Nov 30 2012
From Antti Karttunen, Feb 10 2016: (Start)
a(0) = 1, and for n > 1, a(n) = A048720(3, a(n-1)) = A048724(a(n-1)).
a(n) = A048723(3,n).
a(n) = A193231(A000079(n)).
For all n >= 0: A268389(a(n)) = n.
(End)

A045544 Odd values of n for which a regular n-gon can be constructed by compass and straightedge.

Original entry on oeis.org

3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295

Offset: 1

Ken Takusagawa

If there are no more Fermat primes, then 4294967295 is the last term in the sequence.
From Daniel Forgues, Jun 17 2011: (Start)
The 31 = 2^5 - 1 terms of this sequence are the nonempty products of distinct Fermat primes. The 5 known Fermat primes are in A019434.
Prepending the empty product, i.e., 1, to this sequence gives A004729.
The initial term for this sequence is thus a(1) (offset=1), since a(0) should correspond to the omitted empty product, term a(0) of A004729.
Rows 1 to 31 of Sierpiński's triangle, if interpreted as a binary number converted to decimal (A001317), give a(1) to a(31). (End)

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 73 at pp. 181-182.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.
OEIS Wiki, Constructible odd-sided polygons.
OEIS Wiki, Sierpinski's triangle.

Cf. A019434. Essentially same as A004729.
Coincides with A001317 for the first 31 terms only. - Robert G. Wilson v, Dec 22 2001
Cf. A053576.

Union[Times@@@Rest[Subsets[{3,5,17,257,65537}]]] (* Harvey P. Dale, Sep 20 2011 *)

Each term is the product of distinct odd Fermat primes.

Sum_{n>=1} 1/a(n) = -1 + Product_{n>=1} (1+1/A019434(n)) = 0.7007354948... >= 1003212011/1431655765 = sigma(2^32-1)/(2^32-1) - 1, with equality if there are only five Fermat primes (A019434). - Amiram Eldar, Jan 22 2022

A078164 Numbers k such that phi(k) is a perfect biquadrate.

Original entry on oeis.org

1, 2, 17, 32, 34, 40, 48, 60, 257, 512, 514, 544, 640, 680, 768, 816, 960, 1020, 1297, 1387, 1417, 1729, 1971, 2109, 2223, 2289, 2331, 2445, 2457, 2565, 2594, 2608, 2774, 2812, 2834, 2835, 3052, 3260, 3458, 3888, 3912, 3924, 3942, 3996, 4104, 4212, 4218

Offset: 1

Labos Elemer, Nov 27 2002

nonn

Corresponding values of phi include 1, 16, 256, 1296, 4096, ... and these arise several times each.
a(3) = A053576(4).
A013776 is a subsequence since phi(2^(4*n+1)) = (2^n)^4. - Bernard Schott, Sep 22 2022
Subsequence of primes is A037896 since in this case: phi(k^4+1) = k^4. - Bernard Schott, Mar 05 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A039770. A037896 is a subsequence.
Sequences where phi(k) is a perfect power: A039770 (square), A039771 (cube), this sequence (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th).
Cf. A001317, A053576, A037896, A045544, A000010, A013776.

k=4; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Select[Range[5000],IntegerQ[Surd[EulerPhi[#],4]]&] (* Harvey P. Dale, Apr 30 2015 *)

is(n)=ispower(eulerphi(n),4) \\ Charles R Greathouse IV, Apr 24 2020

from itertools import count, islice
from sympy import totient, integer_nthroot
def A078164_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:integer_nthroot(totient(n),4)[1], count(max(1,startvalue)))
A078164_list = list(islice(A078164_gen(),20)) # Chai Wah Wu, Feb 28 2023

A004729 Divisors of 2^32 - 1 (for a(1) to a(31), the 31 regular polygons with an odd number of sides constructible with ruler and compass).

Original entry on oeis.org

Offset: 0

N. J. A. Sloane

The 32 divisors of the product of the 5 known Fermat primes.
The only known odd numbers whose totient is a power of 2. - Labos Elemer, Dec 06 2000
Equals first 32 members of A001317. Also, equals first 32 members of A053576. - Omar E. Pol, Dec 10 2008
Omitting the first term a(0)=1 gives A045544 (the number of sides of constructible odd-sided regular polygons.)

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 140.

Eric Weisstein's World of Mathematics, Regular Polygon, Sierpiński Sieve, Constructible Polygon
OEIS Wiki, Constructible odd-sided polygons
OEIS Wiki, Sierpinski's triangle
Index entries for sequences related to divisors of numbers

Essentially same as A045544.

Cf. A000010, A000215, A001317, A003401, A003527, A004169, A004729, A019434, A045544, A047999, A053576, A054432.

Mathematica
```
Divisors[2^32-1]
```
PARI
```
divisors(1<<32-1)
```

Edited by Daniel Forgues, Jun 17 2011

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

Original entry on oeis.org

0, 1, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Labos Elemer, Apr 28 2003

a(n) is the least number x such that gcd(2^x, x-phi(x)) = 2^n. If cototient is replaced by totient, analogous values are different: A053576.

			G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000010, A007283, A009195, A050339, A134059.

Essentially the same as A003945 (and perhaps also A058764).

[0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // Klaus Brockhaus, Dec 02 2009

0,1,seq(3*2^(n-1), n=2..40); # G. C. Greubel, Apr 27 2021

{0}~Join~Map[Total, {{1}}~Join~Table[3 Binomial[n, k], {n, 30}, {k, 0, n}]] (* Michael De Vlieger, Jul 03 2016, after Harvey P. Dale at A134059 *)
Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* G. C. Greubel, Apr 27 2021 *)
Join[{0,1},NestList[2#&,6,30]] (* Harvey P. Dale, Jan 22 2024 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-6*k + 16) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n)=if(n<2,n,3<<(n-1)) \\ Charles R Greathouse IV, Jun 16 2012

[0,1]+[3*2^(n-1) for n in (2..40)] # G. C. Greubel, Apr 27 2021

a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.
G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - Michael Somos, Jun 15 2012
Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - Gary W. Adamson, Nov 18 2007
a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - Philippe Deléham, Oct 30 2008
a(n) = (-6*n + 16) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - Ilya Gutkovskiy, Jul 04 2016
a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - G. C. Greubel, Apr 27 2021

More terms from Klaus Brockhaus, Dec 02 2009

A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 641, 6700417, 274177, 67280421310721, 59649589127497217, 5704689200685129054721, 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321

Offset: 0

N. J. A. Sloane, Dec 30 1999

Alternatively, list of prime factors of terms of A001317 in order of their first appearance. - Labos Elemer, Jan 21 2002
From T. D. Noe, Jan 29 2009: (Start)
That these two definitions give the same sequence follows from the fact (stated as a formula in A001317) that A001317(n) is the product of Fermat numbers F(i) according to which bits of n are set.
For instance, for n=41, the binary representation of n is 101001, which has bits 0, 3 and 5 set. A001317(n) = 3311419785987 = 3*257*4294967297 = F(0)*F(3)*F(5).
This factorization also explains why the "first 31 numbers give odd-sided constructible polygons". I think Hewgill first noticed this factorization. (End)

			Triangle begins:
  3;
  5;
  17;
  257;
  65537;
  641,               6700417;
  274177,            67280421310721;
  59649589127497217, 5704689200685129054721;
  1238926361552897,  93461639715357977769163558199606896584051237541638188580280321;
  ...
A001317(127) = 3*5*17*257*65537.641*6700417*274177*6728042130721, A001317(128) = 59649589127497217*5704689200685129054721. See also A050922. Compare with A053576, where 2 and A000215 appear as prime factors. - _Labos Elemer_, Jan 21 2002

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..29
J. Bernheiden, Fermat Numbers (Text in German)
R. P. Brent, Factorization of the tenth Fermat number
R. P. Brent, Factorization of the eleventh Fermat number
R. P. Brent, Succinct proofs of primality for the factors of some Fermat numbers
R. P. Brent & J. M. Pollard, Factorization of the eighth Fermat number
R. P. Brent et al., Three new factors of Fermat numbers
C. K. Caldwell, The Prime Glossary, Fermat divisor
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012. - From _N. J. A. Sloane_, Jun 13 2012
R. Munafo, Notes on Fermat numbers
Mercedes Orús-Lacort, Fermat numbers are not prime numbers for n >= 5, (2020).
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A019434, A023394, A093179.

Cf. A001317, A001316, A003401, A045544, A053576.

Flatten[Transpose[FactorInteger[#]][[1]]&/@Table[2^(2^n)+1,{n,0,8}]] (* Harvey P. Dale, May 18 2012 *)

for(n=0, 1e3, f=factor(2^(2^n)+1)[, 1]; for(i=1, #f, print1(f[i], ", "))) \\ Felix Fröhlich, Aug 16 2014

More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000.
Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of T. D. Noe
Link to Munafo webpage fixed by Robert Munafo, Dec 09 2009

cp's OEIS Frontend

A001317 Sierpiński's triangle (Pascal's triangle mod 2) converted to decimal.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

Python

Python

Formula

A045544 Odd values of n for which a regular n-gon can be constructed by compass and straightedge.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A078164 Numbers k such that phi(k) is a perfect biquadrate.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

A004729 Divisors of 2^32 - 1 (for a(1) to a(31), the 31 regular polygons with an odd number of sides constructible with ruler and compass).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.

Views

Author

Keywords

Comments

Examples

References