A000225 -id:A000225 - cp's OEIS frontend

A227843 Numbers n such that the binary XOR of the divisors of n (A178910(n)) is a binary repunit (A000225).

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 2592, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 2458624, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 1

Alex Ratushnyak, Jul 06 2013

These are also numbers n such that A178910(n) >= A178910(i) for all i

All powers of 2 are in the sequence. Terms that are not 2^x are 2592 and 2458624. No other non-2^x terms below 2^35.

Cf. A000225, A178910.

fQ[n_] := Union@ IntegerDigits[ Fold[ BitXor[#1, #2] &, 0, Divisors@ n], 2] == {1}; Select[ Range@ 1000000000, fQ] (* Robert G. Wilson v, Aug 22 2013 *)

A247092 Limiting sequence obtained by taking the sequence of Mersenne numbers 2^n-1, n=1,2,...(A000225) and applying an infinite process which is described in the comments.

Original entry on oeis.org

1, 0, 4, 2, 0, 32, 8, 1, 0, 64, 4, 0, 1024, 32, 0, 32768, 512, 4, 0, 16384, 64, 0, 1048576, 2048, 2, 0, 131072, 64, 0, 16777216, 4096, 0, 4294967296, 524288, 32, 0, 134217728, 4096, 0, 68719476736, 1048576, 8, 0, 536870912, 2048, 0, 549755813888, 1048576, 1, 0

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Nov 18 2014

Write the Mersenne numbers 2^n-1, n=1,2,..., in binary in form of triangle T_0(M) consisting of all 1's:
1
11
111
1111
11111
......
Let the operator A_k map every k-th entry to its binary opposite (1->0, 0->1), for k=2,3,... . Put T_inf(M) = ...*A_4*A_3*A_2(T_(0)M), with successive applications of the operators A_2, A_3, A_4, ...
Note that the (0,1)-triangle T_inf(M) is well-defined, since the operator T_n does not affect entries in the first floor((sqrt(8*n-7) - 1)/2) rows.
The sequence lists numbers obtained by reading rows of T_inf(M) in binary and converting them to decimal.
The n-th entry t_n of T_inf(M) equals 1, if n is perfect square, and 0 otherwise (A010052, for n>=1).
Indeed, in order to get an entry t_n of T_inf(M), we should use all considered operators A_d, d|n, d>1. The number of these operators is the diminished on 1 the number of divisors of n which is even iff n is a perfect square. Thus only in this case we obtain that entry in the n-th position is flipped, beginning with 1, an even number of times, such that t_n=1, while, if n is nonsquare, t_n=0.
Note that, since (n+1)^2 - n^2 > n, then in every row of T_inf(M) there exists at most one 1. So every term is either 0 or a power of 2.

			T_inf(M) begins
1
00
100
0010
00000
100000
0001000
00000001
.........
Let n=4. Then the interval in the formula is [sqrt(7), sqrt(10)], so x=3 and a(4) = 2^(10-9) = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..3453 (all terms less than 10^1000)

Cf. A000225, A010052, A246553.

a(n)=my(mx=n*(n+1)/2,s=sqrtint(mx)^2); if(s<=mx-n, 0, 2^(mx-s)) \\ Charles R Greathouse IV, Nov 19 2014

If there is an integer x in [sqrt((n-1)*n/2 +1), sqrt(n*(n+1)/2)] then it is unique and a(n) = 2^(n(n+1)/2-x^2); otherwise, a(n)=0.

Thus there are n/sqrt(2) + O(1) positive terms among the first n.

a(25)-a(50) from Charles R Greathouse IV, Nov 19 2014

A260757 Least k > 0 such that M(n)^2 - 2k is prime, where M(n) = 2^n - 1 = A000225(n).

Original entry on oeis.org

1, 2, 1, 1, 1, 4, 1, 1, 7, 10, 1, 10, 1, 10, 5, 1, 14, 24, 1, 1, 13, 1, 16, 3, 82, 1, 19, 1, 23, 94, 64, 58, 7, 6, 14, 3, 46, 22, 5, 13, 107, 69, 38, 90, 59, 75, 104, 25, 4, 10, 14, 4, 44, 10, 5, 1, 77, 81, 85, 94, 71, 9, 14, 111, 13, 27, 20, 9, 37, 6, 5, 4, 62, 12, 38, 4, 37

Offset: 0

M. F. Hasler, Jul 30 2015

nonn

For n = 0 and n = 1, no k > 0 can yield a positive prime, the given values are the smallest to yield the opposite of a positive prime: M(0)^2 - 2*1 = 0 - 2 = -2 and M(1)^2 - 2*2 = 1 - 4 = -3.

			For n = 2, M(2) = 2^2 - 1 = 3 and 3*3 - 2k = 7 is a prime for k=1, thus a(2) = 1.
For n = 3, M(3) = 2^3 - 1 = 7 and 7*7 - 2k = 47 is a prime for k=1, thus a(3) = 1.
For n = 4, M(4) = 2^4 - 1 = 15 and 15*15 - 2k = 223 is a prime for k=1, thus a(4) = 1.
For n = 5, M(5) = 2^5 - 1 = 31 and 31*31 - 2k = 953 is prime for k=4 and no smaller k, thus a(5) = 4.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A091515 (a(n)=1 for n > 0), A260758.

f:= proc(n) local r;
  r:= (2^n-1)^2;
  (r - prevprime(r))/2
end proc:
f(0):=1: f(1):= 2:
map(f, [$0..100]); # Robert Israel, Apr 02 2020

f[n_] := Module[{r = (2^n - 1)^2}, (r - NextPrime[r, -1])/2 ];
f[0] = 1; f[1] = 2;
f /@ Range[0, 100] (* Jean-François Alcover, Jul 28 2020, after Robert Israel *)

a(n)={n>1&&for(k=1,9e9,ispseudoprime((2^n-1)^2-2*k)&&return(k));n+1}

a(n) = 1 for n=0 or n in A091515.

A087100 A000225 (2^n - 1) interlaced with A008593 (11n).

Original entry on oeis.org

0, 1, 11, 3, 22, 7, 33, 15, 44, 31, 55, 63, 66, 127, 77, 255, 88, 511, 99, 1023, 110, 2047, 121, 4095, 132, 8191, 143, 16383, 154, 32767, 165, 65535, 176, 131071, 187, 262143, 198, 524287, 209, 1048575, 220, 2097151, 231, 4194303, 242, 8388607, 253

Offset: 0

Jeremy Gardiner, Aug 10 2003

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-5,0,2).

Cf. A063914, A087098, A086848.

Rest[With[{nn=30},Riffle[2^Range[0,nn]-1,11Range[0,nn]]]] (* Harvey P. Dale, Aug 15 2011 *)

From Chai Wah Wu, Feb 02 2021: (Start)
a(n) = 4*a(n-2) - 5*a(n-4) + 2*a(n-6) for n > 5.
G.f.: x*(22*x^3 + x^2 - 11*x - 1)/((x - 1)^2*(x + 1)^2*(2*x^2 - 1)). (End)

A140407 A000225 interleaved with A000051.

Original entry on oeis.org

1, 2, 3, 3, 7, 5, 15, 9, 31, 17, 63, 33, 127, 65, 255, 129, 511, 257, 1023, 513, 2047, 1025, 4095, 2049, 8191, 4097, 16383, 8193, 32767, 16385, 65535, 32769, 131071, 65537, 262143, 131073, 524287, 262145, 1048575, 524289, 2097151, 1048577, 4194303

Offset: 0

Paul Curtz, Jun 16 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,2,2).

Cf. A000051, A000225, A000079, A135530.

LinearRecurrence[{-1,2,2},{1,2,3},50] (* Harvey P. Dale, Apr 03 2013 *)

def A140407(n): return 2 if n == 1 else (1<<(n>>1))|1 if n&1 else -1^(-2<<(n>>1)) # Chai Wah Wu, Dec 21 2022

a(2n) = A000225(n+1) = A135530(2n) - 1. a(2n+1) = A000051(n) = 1 + A135530(2n+1).
a(n) = -a(n-1) + 2*a(n-2) + 2*a(n-3). a(2n) + a(2n+1) = 3*A000079(n).
O.g.f.: (1 + 3x + 3x^2)/((1+x)*(1-2x^2)). - R. J. Mathar, Jul 08 2008

Edited and extended by R. J. Mathar, Jul 08 2008

A140745 Smallest prime p such that the Mersenne number A000225(p) = 2^p - 1 has exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

2, 11, 29, 157, 113, 223, 491, 431, 397

Offset: 1

Lekraj Beedassy, Jul 12 2008

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 223, pp 63-4, Ellipse Paris 2008.

Cf. A000225, A001222, A046051, A136033.

a[n_]:=Module[{p=0},Until[PrimeOmega[2^Prime[p]-1]==n,p++];Prime[p]];Array[a,6] (* James C. McMahon, Jul 14 2025 *)

a(n) = forprime(p=2, oo, if(bigomega(2^p-1)==n, return(p))); \\ Jinyuan Wang, Aug 10 2021

A140769 Numbers n such that consecutive Mersenne numbers A000225(n-1) = M(n-1) = 2^(n-1) - 1 and A000225(n) = M(n) = 2^n - 1 are both squarefree.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 23, 26, 27, 28, 29, 32, 33, 34, 35, 38, 39, 44, 45, 46, 47, 50, 51, 52, 53, 56, 57, 58, 59, 62, 65, 68, 69, 70, 71, 74, 75, 76, 77, 82, 83, 86, 87, 88, 89, 92, 93, 94, 95, 98, 99, 104, 107, 112, 113, 116, 117, 118, 119, 122, 123, 124

Offset: 1

Lekraj Beedassy, Jul 13 2008

nonn

J.-M. De Koninck, Ces nombres qui nous fascinent, Footnote pp 18, Ellipses, Paris 2008.

Amiram Eldar, Table of n, a(n) for n = 1..629

Cf. A000225, A005117.

A221975 Triangle read by rows of the numbers that are the sum of some consecutive Mersenne numbers A000225.

Original entry on oeis.org

1, 3, 4, 7, 10, 11, 15, 22, 25, 26, 31, 46, 53, 56, 57, 63, 94, 109, 116, 119, 120, 127, 190, 221, 236, 243, 246, 247, 255, 382, 445, 476, 491, 498, 501, 502, 511, 766, 893, 956, 987, 1002, 1009, 1012, 1013, 1023, 1534, 1789, 1916, 1979, 2010, 2025, 2032, 2035, 2036

Offset: 1

Omar E. Pol, Feb 12 2013

			Triangle begins:
1;
3,     4;
7,    10,  11;
15,   22,  25,  26;
31,   46,  53,  56,  57;
63,   94, 109, 116, 119, 120;
127, 190, 221, 236, 243, 246, 247;
255, 382, 445, 476, 491, 498, 501, 502;
...

Column 1 is A000225. Right border gives the positive terms of A000295.

Cf. A023758, A141419, A194939.

T(n,k) = sum_{j = n-k+1..n} 2^j - 1, n>=1, k>=1.

T(n,n) = A000295(n+1).

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cp's OEIS Frontend

A086652 a(n) = A000225(n+3)-A052955(n).

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A224520 Numbers a(n) with property a(n) + a(n+4) = 2^(n+4) - 1 = A000225(n+4).

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A227843 Numbers n such that the binary XOR of the divisors of n (A178910(n)) is a binary repunit (A000225).

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A247092 Limiting sequence obtained by taking the sequence of Mersenne numbers 2^n-1, n=1,2,...(A000225) and applying an infinite process which is described in the comments.

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A260757 Least k > 0 such that M(n)^2 - 2k is prime, where M(n) = 2^n - 1 = A000225(n).

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A087100 A000225 (2^n - 1) interlaced with A008593 (11n).

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A140407 A000225 interleaved with A000051.

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A140745 Smallest prime p such that the Mersenne number A000225(p) = 2^p - 1 has exactly n prime factors (counted with multiplicity).

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