A001146 -id:A001146 - cp's OEIS frontend

A242323 Number of binary words of length n that contain all 32 5-bit words as (possibly overlapping) contiguous subwords.

65536, 352256, 1442816, 5313536, 18323520, 60481632, 192562808, 593792608, 1782459992, 5221699004, 14967607810, 42060446246, 116067269324

Offset: 36

Alois P. Heinz, May 10 2014

			a(36) = 65536: 000001000110010100111010110111110000, ... .

Eric Weisstein's World of Mathematics, Coin Tossing

Cf. A001146, A052944, A242167, A242206, A242257.

b:= proc(n, t, s) option remember; `if`(s={}, 2^n,
      `if`(nops(s)>n, 0, b(n-1, irem(2*t, 16), s minus {2*t})
        +b(n-1, irem(2*t+1, 16), s minus {2*t+1})))
    end:
a:= n-> add(b(n-4, j, {$0..31}), j=0..15):
seq(a(n), n=36..37);

b[n_, t_, s_] := b[n, t, s] = If[s == {}, 2^n,
   If[Length[s] > n, 0, b[n-1, Mod[2*t, 16], s~Complement~{2*t}] +
   b[n-1, Mod[2*t+1, 16], s~Complement~{2*t+1}]]];
a[n_] := Sum[b[n-4, j, Range[0, 31]], {j, 0, 15}];
Table[a[n], {n, 36, 39}] (* Jean-François Alcover, Sep 06 2022, after Alois P. Heinz *)

a(44)-a(48) from Alois P. Heinz, Feb 27 2015

A318130 Number of sets of subsets of {1,...,n} with intersection {}.

Original entry on oeis.org

2, 3, 11, 219, 64595, 4294642035, 18446744047940725979, 340282366920938463334247399005993378251, 115792089237316195423570985008687907850547725730273056332267095982282337798563

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			The a(2) = 11 sets of sets:
  {}
  {{}}
  {{},{1}}
  {{},{2}}
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Cf. A000371, A001146, A003465, A119563, A131288, A318128, A318129.

Cf. A318128, A318129, A318131, A318132.

Table[Length[Select[Subsets[Subsets[Range[n]]],Or[#=={},Intersection@@#=={}]&]],{n,0,4}]

Binomial transform of A131288.

Inverse binomial transform of A119563(n) = 2^(2^n) + 2^n - 1.

A336066 Numbers k such that the exponent of the highest power of 2 dividing k (A007814) is a divisor of k.

Original entry on oeis.org

2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 90, 92, 94, 98, 100, 102, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 138, 140, 142, 144

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

All the terms are even by definition.
If m is a term then m*(2*k+1) is a term for all k>=1.
Šalát (1994) proved that the asymptotic density of this sequence is 0.435611... (A336067).

			2 is a term since A007814(2) = 1 is a divisor of 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tibor Šalát, On the function a_p, p^a_p(n) || n (n > 1), Mathematica Slovaca, Vol. 44, No. 2 (1994), pp. 143-151.

A001146 and A039956 are subsequences.
Cf. A007814, A336067.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336068.

Select[Range[2, 150, 2], Divisible[#, IntegerExponent[#, 2]] &]

isok(m) = if (!(m%2), (m % valuation(m,2)) == 0); \\ Michel Marcus, Jul 08 2020

from itertools import count, islice
def A336066_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:n%(~n&n-1).bit_length()==0,count(max(startvalue+startvalue&1,2),2))
A336066_list = list(islice(A336066_gen(startvalue=3),30)) # Chai Wah Wu, Jul 10 2022

A001042 a(n) = a(n-1)^2 - a(n-2)^2.

Original entry on oeis.org

1, 2, 3, 5, 16, 231, 53105, 2820087664, 7952894429824835871, 63248529811938901240357985099443351745, 4000376523371723941902615329287219027543200136435757892789536976747706216384

Offset: 0

N. J. A. Sloane, R. K. Guy

The next term has 152 digits. - Franklin T. Adams-Watters, Jun 11 2009

Archimedeans Problems Drive, Eureka, 27 (1964), 6.
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).

Reinhard Zumkeller, Table of n, a(n) for n = 0..13
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001146, A000278, A000283, A058182, A007018.

Cf. A064236 (numbers of digits).

a001042 n = a001042_list !! n
a001042_list = 1 : 2 : zipWith (-) (tail xs) xs
               where xs = map (^ 2) a001042_list
-- Reinhard Zumkeller, Dec 16 2013

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]^2-a[n-2]^2},a,{n,0,12}] (* Harvey P. Dale, Jan 11 2013 *)

a(n) ~ c^(2^n), where c = 1.1853051643868354640833201434870139866230288004895868726506278977814490371... . - Vaclav Kotesovec, Dec 17 2014

More terms from James Sellers, Sep 19 2000.

A057755 Number of digits in n-th Fermat number (A000215).

Original entry on oeis.org

1, 1, 2, 3, 5, 10, 20, 39, 78, 155, 309, 617, 1234, 2467, 4933, 9865, 19729, 39457, 78914, 157827, 315653, 631306, 1262612, 2525223, 5050446, 10100891, 20201782, 40403563, 80807125, 161614249, 323228497, 646456994, 1292913987, 2585827973

Offset: 0

Robert G. Wilson v, Oct 30 2000

Also number of digits of A001146(n) and A051179(n). - Michel Marcus, Dec 21 2018

			a(6) = 20 because 2^(2^6) + 1 = 18446744073709551617 which is a twenty-digit number.

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1995, page 139.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 100 terms from Jinyuan Wang)
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 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A001146, A051179.

List([0..18],n->Size(ListOfDigits(2^(2^n)+1))); # Muniru A Asiru, Dec 20 2018

[Floor(2^n*Log(10,2)/Log(10,10))+1: n in [0..40]]; // Vincenzo Librandi, Nov 08 2018

seq(length(2^(2^n)),n=0..20); # Zerinvary Lajos, Apr 20 2008

Table[ Floor[ 2^n * N[ Log[ 10, 2 ], 24 ] + 1 ], {n, 0, 43} ]

for(n=0, 50, print(n, " ", floor(2^n*log(2)/log(10))+1); ) \\ Jinyuan Wang, Nov 07 2018

a(n) = floor(log_10(F_n)+1) (F_n is the n-th Fermat number). - Ivan Panchenko, Sep 06 2009

A073113 a(n) = 2^(2^n + n).

Original entry on oeis.org

2, 8, 64, 2048, 1048576, 137438953472, 1180591620717411303424, 43556142965880123323311949751266331066368

Offset: 0

Joe Mathes (oldschoolchaz(AT)hotmail.com), Aug 19 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 0..11

Cf. A000079, A001146, A006127.

a[n_] := 2^(2^n + n); Array[a, 8, 0] (* Amiram Eldar, Aug 14 2022 *)

a(n) = 2^A006127(n) = A000079(n)*A001146(n). - Amiram Eldar, Aug 14 2022

A181794 Numbers n such that the number of even divisors of n is an even divisor of n.

Original entry on oeis.org

4, 6, 10, 12, 14, 16, 20, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 52, 58, 62, 68, 74, 76, 80, 82, 86, 90, 92, 94, 106, 112, 116, 118, 120, 122, 124, 126, 134, 142, 144, 146, 148, 150, 158, 160, 164, 166, 168, 172, 176, 178, 180, 188, 192, 194, 198, 202, 206, 208, 212, 214, 216, 218, 226, 234, 236, 240, 244, 252, 254, 256, 262, 264, 268, 272, 274, 278

Offset: 1

Matthew Vandermast, Nov 14 2010

nonn

All terms are even, since odd numbers, even if they have an even count of divisors, don't have any even divisors.

Includes all numbers of the form A000040(m)*A001146(n).

			a(4)=12 has four even divisors (2, 4, 6, and 12), and 4 is one of those even divisors.
The number 21 is not in this sequence: it has four divisors (1, 3, 7, and 21), and 4 is not one of those divisors.

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

A100484 and A001749 are subsequences. A001146 and A100042 are also subsequences except for their initial terms.

A183169 Tree generated by the squares.

Original entry on oeis.org

1, 2, 4, 3, 16, 6, 9, 5, 256, 20, 36, 8, 81, 12, 25, 7, 65536, 272, 400, 24, 1296, 42, 64, 11, 6561, 90, 144, 15, 625, 30, 49, 10, 4294967296, 65792, 73984, 288, 160000, 420, 576, 29, 1679616, 1332, 1764, 48, 4096, 72, 121, 14

Offset: 1

Clark Kimberling, Dec 28 2010

A permutation of the positive integers. See the comment at A183079 (tree generated by the triangular numbers). The leftmost numbers (1,2,4,16,...) are, after the initial 1, given by A001146. The rightmost numbers (1,2,3,5,7,10,...) are, after the initial 1, the iterates of the nonsquare function; see a comment at A033638.

			First levels of the tree:
......................1
......................2
...........4.....................3
.......16.......6...........9..........5
...256...20...36..8......81...12....25...7

Cf. A183079, A000037, A183169, A001146, A033638.

Let L(n) be the n-th square (A000290).
Let U(n) be the n-th nonsquare (A000037).
The tree-array T(n,k) is then given by rows:
T(0,0)=1; T(1,0)=2;
T(n,2j)=L(T(n-1),j);
T(n,2j+1)=U(T(n-1),j);
for j=0,1,...,2^(n-1)-1, n>=2.

A211667 Number of iterations sqrt(sqrt(sqrt(...(n)...))) such that the result is < 2.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Hieronymus Fischer, Apr 30 2012

Different from A001069, but equal for n < 256.

			a(n) = 1, 2, 3, 4, 5, ... for n = 2^1, 2^2, 2^4, 2^8, 2^16, ..., i.e., n = 2, 4, 16, 256, 65536, ... = A001146.

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

Cf. A001069, A001146, A010096, A211662, A211668, A211670.

Cf. A087046 (run lengths).

a[n_] := Length[NestWhileList[Sqrt, n, # >= 2 &]] - 1; Array[a, 100] (* Amiram Eldar, Dec 08 2018 *)

apply( A211667(n, c=0)={while(n>=2, n=sqrtint(n); c++); c}, [1..50]) \\ This defines the function A211667. The apply(...) provides a check and illustration. - M. F. Hasler, Dec 07 2018

a(n) = if(n<=1,0, logint(logint(n,2),2) + 1); \\ Kevin Ryde, Jan 18 2024

A211667=lambda n: n and (n.bit_length()-1).bit_length() # Natalia L. Skirrow, May 16 2023

a(2^(2^n)) = a(2^(2^(n-1))) + 1, for n >= 1.
G.f.: g(x) = 1/(1-x)*Sum_{k>=0} x^(2^(2^k))
  = (x^2 + x^4 + x^16 + x^256 + x^65536 + ...)/(1 - x).
a(n) = 1 + floor(log_2(log_2(n))) for n>=2. - Kevin Ryde, Jan 18 2024

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 105, 255, 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, 98304, 245760, 417792, 430080, 1044480, 1582080, 3947520, 3982080, 6908160, 6919680, 16776960, 106991625, 267448335, 1019462460, 1771476585, 4294967295

Offset: 0

Tilman Piesk, Aug 01 2013

A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).
Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:
1,
3,
6, 15,
24, 60, 105, 255,
384, 960, 1632, 1680, 4080, 15555, 27030, 65535...
The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).
The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7,  correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.
Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)
a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

Tilman Piesk, Table of n, a(n) for n = 0..147
Tilman Piesk, a(n) for n = 0..147 and corresponding entries of A190939 in reverse binary. Prime factors and numbers of prime factors (A001222).
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

Subsequence of A227723 (all becs). All entries are also in A227963 (all sona-secs). Neither shares the property of divisibility by 3.
A190939, A034343, A076766, A076831.
The prime factors contain many prime factors of Fermat numbers (A023394).

a( A076766 - 1 ) = A001146 - 1 = A051179.

a( A076766 ) = A001146 * 3/2 (probably).

cp's OEIS Frontend

A242323 Number of binary words of length n that contain all 32 5-bit words as (possibly overlapping) contiguous subwords.

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A318130 Number of sets of subsets of {1,...,n} with intersection {}.

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A336066 Numbers k such that the exponent of the highest power of 2 dividing k (A007814) is a divisor of k.

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A001042 a(n) = a(n-1)^2 - a(n-2)^2.

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A057755 Number of digits in n-th Fermat number (A000215).

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A073113 a(n) = 2^(2^n + n).

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A181794 Numbers n such that the number of even divisors of n is an even divisor of n.

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