A116360 -id:A116360 - cp's OEIS frontend

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 17, 18, 19, 24, 25, 32, 33, 34, 35, 36, 38, 48, 49, 50, 51, 64, 65, 66, 67, 68, 70, 72, 73, 76, 96, 97, 98, 99, 100, 102, 128, 129, 130, 131, 132, 134, 136, 137, 140, 144, 145, 146, 147, 152, 153, 192, 193, 194, 195, 196, 198, 200, 201

Offset: 1

Antti Karttunen, Mar 30 1999

If bit i is 1, then bits i+-2 must be 0. All terms satisfy A048725(n) = 5*n.
It appears that n is in the sequence if and only if C(5n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
Yes, as remarked in A048715, "This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p." - Jason Kimberley, Dec 21 2011
A116361(a(n)) <= 2. - Reinhard Zumkeller, Feb 04 2006

Superset of A048715 and A048719. Union of A004742 and A003726.

Cf. A048729, A003714, A115845, A115847, A116360.

Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{_, 0, , 1, __}|{_ 1, , 0, __}] && !MatchQ[bb, {_, 1, , 1, __}]];
Select[Range[0, 201], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n,n>>2) \\ Charles R Greathouse IV, Oct 03 2016

list(lim)=my(v=List(),n,t); while(n<=lim, t=bitand(n,n>>2); if(t, n+=1<Charles R Greathouse IV, Oct 22 2021

A115847 Integers i such that 17i = 17 X i, i.e., 16i XOR i = 17*i.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 36, 37, 40, 41, 44, 45, 48, 52, 56, 60, 64, 65, 66, 67, 72, 73, 74, 75, 80, 82, 88, 90, 96, 97, 104, 105, 112, 120, 128, 129, 130, 131, 132, 133, 134, 135, 144, 146, 148, 150

Offset: 0

Antti Karttunen, Feb 01 2006

nonn

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

A116361(a(n)) <= 4. - Reinhard Zumkeller, Feb 04 2006

Cf. A115848 shows this sequence in binary. Complement of A115849. Differs from A032966 for the first time at n=25, where A032966(25)=34 while a(25)=33.

Cf. A003714, A048716, A115845, A116360.

is(n)=bitxor(n,16*n)==17*n \\ Charles R Greathouse IV, Dec 08 2013

A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17, 20, 21, 24, 28, 32, 33, 34, 35, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 84, 85, 96, 97, 98, 99, 112, 113, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 140, 142, 160, 161, 162, 163, 168, 170, 192

Offset: 1

Antti Karttunen, Feb 01 2006

nonn

Equivalently, numbers n such that 9*n = 9 X n, i.e., 8*n XOR n = 9*n. Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
Equivalently, numbers n such that the binomial coefficient C(9n,n) (A169958) is odd. - Zak Seidov, Aug 06 2010
The equivalence of these three definitions follows from Lucas's theorem on binomial coefficients. - N. J. A. Sloane, Sep 01 2010
Clearly all numbers k*2^i for 1 <= k <= 7 have this property. - N. J. A. Sloane, Sep 01 2010
A116361(a(n)) <= 3. - Reinhard Zumkeller, Feb 04 2006

N. J. A. Sloane and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences defined by congruent products between domains N and GF(2)[X]
Index entries for sequences defined by congruent products under XOR

A115846 shows this sequence in binary.
A033052 is a subsequence.
Cf. A003714, A048716, A115847, A116360, A005809, A003714, A048716, A048715.

Reap[Do[If[OddQ[Binomial[9n,n]],Sow[n]],{n,0,400}]][[2,1]] (* Zak Seidov, Aug 06 2010 *)

is(n)=!bitand(n,n<<3) \\ Charles R Greathouse IV, Sep 23 2012

a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 23 2012

Edited with a new definition by N. J. A. Sloane, Sep 01 2010, merging this sequence with a sequence submitted by Zak Seidov, Aug 06 2010

A116357 Number of partitions of n into products of two successive primes (A006094).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 2, 0, 2, 3, 0, 1, 2, 1, 2, 3, 0, 1, 3, 1, 3, 3, 0, 2, 3, 1, 3, 3, 1, 2, 3, 1, 3, 4, 1, 3, 3, 1, 4, 4, 1, 3, 3, 2, 4, 4, 1, 3, 5

Offset: 1

Reinhard Zumkeller, Feb 12 2006

nonn

a(A116358(n)) = 0; a(A116359(n)) > 0;

a(n) < A101048(n).

			a(41) = #{2*3 + 5*7} = 1;
a(42) = #{2*3+2*3+2*3+2*3+2*3+2*3+2*3, 2*3+2*3+3*5+3*5} = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006094, A101048, A116358, A116359, A116360.

N:= 200: # to get a(1) to a(N)
Primes:= select(isprime,[2,seq(i,i=3..1+floor(sqrt(N)),2)]):
G:= mul(1/(1 - x^(Primes[i]*Primes[i+1])), i=1..nops(Primes)-1):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Dec 09 2016

m = 105; kmax = PrimePi[Sqrt[m]]; Product[1/(1-x^(Prime[k]*Prime[k+1])), {k, 1, kmax}] + O[x]^(m+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Mar 09 2019, after Robert Israel *)

G.f.: Product_{k >= 1} 1/(1 - x^(prime(k)*prime(k+1))). - Robert Israel, Dec 09 2016

A179485 Sums of two successive primes s such that s+-3 are primes.

Original entry on oeis.org

8, 100, 1120, 1220, 1300, 2240, 2380, 2414, 3536, 3634, 4906, 4940, 5566, 5740, 6706, 7240, 8864, 9224, 9394, 10136, 10850, 12040, 12476, 12586, 12920, 13180, 13334, 13754, 14630, 14720, 15134, 16270, 17710, 18430, 18800, 19916, 21014, 21320

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 16 2010

nonn

Intersection of A001043 and A087695. - Robert Israel, Oct 25 2017

			3+5=8,8-3=5(prime),8+3=11(prime),..

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A074924, A074925, A167597, A176984, A176986, A075593, A075594, A116360, A076565

Cf. A001043, A087695.

q:= 2; p:= 3;
count:= 0:
while count < 100 do
  q:= p; p:= nextprime(p);
  s:= q+p;
  if isprime(s-3) and isprime(s+3) then
    count:= count+1; A[count]:= s;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Oct 25 2017

q=3;Select[Table[Prime[n]+Prime[n+1],{n,7!}],PrimeQ[ #-q]&&PrimeQ[ #+q]&]
Select[Total/@Partition[Prime[Range[1400]],2,1],AllTrue[#+{3,-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2018 *)

cp's OEIS Frontend

A048716 Numbers n such that binary expansion matches ((0)*00(1?)1)*(0*).

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A115847 Integers i such that 17*i = 17 X i, i.e., 16*i XOR i = 17*i.

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A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

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A116357 Number of partitions of n into products of two successive primes (A006094).

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A179485 Sums of two successive primes s such that s+-3 are primes.

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Maple

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A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

A115847 Integers i such that 17i = 17 X i, i.e., 16i XOR i = 17*i.