Displaying 1-10 of 14 results found.
10, 101, 1011, 10111, 101111, 10111111, 101111111, 1011111111111, 10111111111111111111, 101111111111111111111111111111111111, 1011111111111111111111111111111111111111
COMMENTS
Primes of the form 10, 101, 1011, 10111,..
EXAMPLE
a(12)=101111111111111111111111111111111111111111111.
MATHEMATICA
FromDigits/@Select[Table[PadRight[{1, 0}, n, 1], {n, 0, 50}], PrimeQ[ FromDigits[ #, 2]]&] (* Harvey P. Dale, Nov 12 2011 *)
a(n) = number of decimal digits of A007505(n).
+20
0
1, 1, 2, 2, 2, 3, 3, 4, 6, 11, 12, 14, 18, 20, 24, 29, 32, 44, 63, 66, 93, 99, 119, 139, 142, 250, 384, 987, 1267, 1546, 2276, 3817, 4486, 5457, 5666, 7734, 7966, 12532, 15470, 21610, 24183, 25795, 26543, 29220, 37217, 46941, 49667, 70671, 124880, 176102, 211335, 219060, 298833, 361552, 370947, 696203, 944108, 1274988, 1833429, 3457035, 3531640, 3580969
a(0) = 1; for n > 0, a(n) = 3*2^(n-1) - 1.
+10
63
1, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943
COMMENTS
Apart from leading term (which should really be 3/2), same as A055010. a(n) = number of partitions Pi of [n+1] (in standard increasing form) such that the permutation Flatten[Pi] avoids the patterns 2-1-3 and 3-1-2. Example: a(3)=11 counts all 15 partitions of [4] except 13/24, 13/2/4 which contain a 2-1-3 and 14/23, 14/2/3 which contain a 3-1-2. Here "standard increasing form" means the entries are increasing in each block and the blocks are arranged in increasing order of their first entries. - David Callan, Jul 22 2008 An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 42, 138, 162, 168, lead to this sequence. For the central square these vectors lead to the companion sequence A003945. - Johannes W. Meijer, Aug 15 2010 The binary representation of a(n) has n+1 digits, where all digits are 1's except digit n-1. For example: a(4) = 23 = 10111 (2). - Omar E. Pol, Dec 02 2012 If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/ A047679, A162909/ A162910, A071766/ A229742, A245325/ A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n >= 0), and the fractions, term by term, are summed at each level n, then the resulting sequence of integers is a(n) + 1/2, apart from leading term (which should be 1/2). - Yosu Yurramendi, May 23 2015 For n >= 2, A083329(n) in binary representation is a string [101..1], also 10 followed with (n-1) 1's. For n >= 3, A036563(n) in binary representation is a string [1..101], also (n-2) 1's followed with 01. Thus A083329(n) is a reflection of the binary representation of A036563(n+1). Example: A083329(5) = 101111 in binary, A036563(6) = 111101 in binary. - Ctibor O. Zizka, Nov 06 2018
FORMULA
a(n) = (3*2^n - 2 + 0^n)/2.
G.f.: (1-x+x^2)/((1-x)*(1-2*x)).
E.g.f.: (3*exp(2*x) - 2*exp(x) + exp(0))/2.
a(0) = 1, a(n) = sum of all previous terms + n. - Amarnath Murthy, Jun 20 2004 a(n) = 3*a(n-1) - 2*a(n-2) for n > 2, a(0)=1, a(1)=2, a(2)=5. - Philippe Deléham, Nov 29 2013 a(n) = (...((((((1)+1)*2+1)*2+1)*2+1)*2+1)...), with n+1 1's, n >= 0.
a(n) = 2*a(n-1) + 1, n >= 2.
a(n) = 2^n + 2^(n-1) - 1, n >= 2. (End)
EXAMPLE
a(0) = (3*2^0 - 2 + 0^0)/2 = 2/2 = 1 (use 0^0=1).
MATHEMATICA
a[1] = 2; a[n_] := 2a[n - 1] + 1; Table[ a[n], {n, 31}] (* Robert G. Wilson v, May 04 2004 *) Join[{1}, LinearRecurrence[{3, -2}, {2, 5}, 40]] (* Vincenzo Librandi, Jan 01 2016 *)
PROG
(Haskell)
a083329 n = a083329_list !! n
a083329_list = 1 : iterate ((+ 1) . (* 2)) 2
CROSSREFS
Cf. A266550 (independence number of the n-Mycielski graph).
a(0) = 0; for n > 0, a(n) = 3*2^(n-1) - 1.
+10
42
0, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887
COMMENTS
Apart from leading term (which should really be 3/2), same as A083329. Written in binary, a(n) is 1011111...1.
The sequence 2, 5, 11, 23, 47, 95, ... apparently gives values of n such that Nim-factorial(n) = 2. Cf. A059970. However, compare A060152. More work is needed! - John W. Layman, Mar 09 2001 With offset 1, number of (132,3412)-avoiding two-stack sortable permutations.
Number of descents after n+1 iterations of morphism A007413. Let A be the Hessenberg matrix of order n, defined by: A[1,j]=[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-1). - Milan Janjic, Jan 24 2010 a(n) is the total number of records over all length n binary words. A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all i<j. - Geoffrey Critzer, Jul 18 2020 Called Thabit numbers after the Syrian mathematician Thābit ibn Qurra (826 or 836 - 901). - Amiram Eldar, Jun 08 2021
EXAMPLE
a(3) = 3*2^2 - 1 = 3*4 - 1 = 11.
PROG
(Sage) [0]+[3*2^(n-1)-1 for n in (1..35)] # G. C. Greubel, May 06 2019 (GAP) Concatenation([0], List([1..35], n-> 3*2^(n-1)-1)) # G. C. Greubel, May 06 2019
CROSSREFS
Cf. A266550 (independence number of the n-Mycielski graph). Cf. A000079, A000225, A007283, A007413, A010036, A030130, A033484, A059970, A060152, A099258, A118654, A164874, A196168.
Numbers m such that 3*2^m - 1 is prime.
(Formerly M0545 N0195)
+10
38
0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760
REFERENCES
H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
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).
CROSSREFS
Cf. A000043, A007505, A003307, A046865, A079906, A046866, A001771, A005541, A056725, A046867, A079907.
EXTENSIONS
a(60) = 11484018 from The Prime Pages, from Pierre CAMI, Nov 25 2014 Terms moved from Data to b-file, and more terms added to b-file, by Jeppe Stig Nielsen, Sep 07 2021
Integers m such that sigma(m) + sigma(2*m) = 6*m.
+10
4
10, 44, 184, 752, 12224, 49024, 12580864, 206158168064, 885443715520878608384, 226673591177468092350464, 232113757366000005450563584, 3894222643901120685369075227951104, 1020847100762815390371677078221595082752, 17126972312471518572699356075530215722269540352
COMMENTS
This is the case h = 2 of the h-perfect numbers as defined in the Harborth link.
EXAMPLE
10 is a term since sigma(10) + sigma(20) = 60, that is 6*10.
MATHEMATICA
Select[Range[10^7], DivisorSigma[1, #] + DivisorSigma[1, 2 #] == 6 # &] (* Michael De Vlieger, Feb 04 2017 *)
PROG
(PARI) isok(n, h=2) = sigma(n) + sigma(h*n) == 2*n*(h+1);
Fixed points of A332221: Numbers k such that A156552(sigma(k)) is equal to k.
+10
3
2, 3, 5, 7, 11, 19, 23, 31, 47, 55, 79, 87, 127, 191, 383, 1279, 5119, 6143, 8191, 20479, 81919, 131071, 524287, 786431, 1310719, 2147483647
COMMENTS
Equally, numbers k such that sigma(k) is equal to A005940(1+k). The primes in this sequence are obtained by subtracting 1 from those terms of A029747 that are one more than a prime. Questions: Are there other composite terms than 55 and 87? Are there other even terms than 2? (All such even terms should also occur in A332218).
MATHEMATICA
Select[Range[10^5], DivisorSigma[1, #] == Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2], q}, Times @@ Flatten[Table[q = Take[p, -i]; Prime[Count[Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}]]] &] (* Michael De Vlieger, Feb 12 2020, after Robert G. Wilson v at A005940 *)
Prime numbers p such that p-1 or p+1 is a number of least prime signature ( A025487).
+10
3
2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, 47, 59, 61, 71, 73, 97, 127, 179, 181, 191, 193, 211, 239, 241, 257, 359, 383, 419, 421, 431, 433, 479, 577, 719, 769, 839, 863, 1151, 1153, 1259, 1297, 1439, 1801, 2161, 2309, 2311, 2521, 2591, 2593, 2879, 3359, 3361
COMMENTS
The corresponding numbers of least prime signature are A344385. 19 is the first prime not in this sequence.
Questions: 1) Is this sequence infinite? 2) Is log(a(n)) = O(log(n)^2)?
EXAMPLE
17 is a term because 17 - 1 = 16 is a number of least prime signature.
MATHEMATICA
{2}~Join~Select[Prime@ Range[2, 900], AnyTrue[# + {-1, 1}, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater] ] == # &] &] (* Michael De Vlieger, May 16 2021 *)
10, 44, 184, 752, 3796, 12224, 49024, 12580864, 60610624, 1091389696, 2936313088, 46672718384, 58082557696, 78857645056, 118480915456, 206158168064, 292776422368, 346109272672, 393960181792
COMMENTS
Is the number of solutions finite? Do solutions to n+k*phi(n)=sigma(n) exist for all values of k? For k=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 the number of solutions I know below 1000000 is {1, 7, 2, 2, 1, 5, 3, 3, 0, 1, 1}). Not more for larger k.
If 3*2^n-1 is prime for n>0, then 2^n(3*2^n-1) belongs to the sequence; therefore this sequence is infinite if the sequence of primes of the form 3*2^n-1 ( A007505) is infinite. - Matthew Vandermast, Jul 31 2004 3796=4.13.73 and 60610624=64.199.4759 do not belong to the class of numbers mentioned above by Vandermast.
EXAMPLE
n=44, phi(n)=20, sigma(44)=1+2+4+11+22+44=84=44+2*20
MATHEMATICA
ta={{0}}; k=2; Do[g=n; If[Equal[n+k*EulerPhi[n], DivisorSigma[1, n]], ta=Append[ta, n]; Print[n]], {n, 1, 182000000}]; {ta, g}
Nonnegative numbers n such that 6*2^n-1 is prime.
+10
1
0, 1, 2, 3, 5, 6, 10, 17, 33, 37, 42, 54, 63, 75, 93, 102, 142, 205, 215, 305, 323, 390, 457, 469, 826, 1273, 3275, 4203, 5133, 7558, 12675, 14897, 18122, 18818, 25689, 26458, 41627, 51386, 71782, 80329, 85686, 88170, 97062
COMMENTS
The associated primes are in A007505.
EXAMPLE
n=0 is in the sequence because 6*2^0-1=5 is prime. n=1 is in the sequence because 6*2^1-1=11 is prime.
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