A047235 -id:A047235 - cp's OEIS frontend

A093719 a(n) = (n mod 2)^(n mod 3).

1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1

Offset: 0

Reinhard Zumkeller, Apr 12 2004

This is a periodic sequence with period 6. The repeating block is 1,1,0,1,0,1. - Michel Dekking, Sep 19 2020

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A000035, A010872, A047235, A047273, A093718.

List([0..120],n->(n mod 2)^(n mod 3)); # Muniru A Asiru, Dec 19 2018

[(n mod 2)^(n mod 3): n in [0..100]]; // Wesley Ivan Hurt, Oct 13 2014

A093719:=n->(n mod 2)^(n mod 3): seq(A093719(n), n=0..40); # Wesley Ivan Hurt, Oct 13 2014

PadRight[{},120,{1,1,0,1,0,1}] (* Harvey P. Dale, Jun 26 2021 *)

A093719(n) = ((n%2)^(n%3)); \\ Antti Karttunen, Dec 19 2018

a(n) = A000035(n)^A010872(n).
a(A047273(n)) = 1, a(A047235(n)) = 0. [Reinhard Zumkeller, Oct 01 2008]
G.f.: -(x^5 + x^3 + x + 1)/(x^6 - 1). - Colin Barker, Apr 01 2013
E.g.f.: (2*cos(sqrt(3)*x/2)*cosh(x/2) + cosh(x))/3 + sinh(x). - Stefano Spezia, Jul 26 2024

A098764 a(n) = 3p - q where p and q are consecutive primes.

Original entry on oeis.org

3, 4, 8, 10, 20, 22, 32, 34, 40, 56, 56, 70, 80, 82, 88, 100, 116, 116, 130, 140, 140, 154, 160, 170, 190, 200, 202, 212, 214, 212, 250, 256, 272, 268, 296, 296, 308, 322, 328, 340, 356, 352, 380, 382, 392, 386, 410, 442, 452, 454, 460, 476, 472, 496, 508, 520

Offset: 1

Giovanni Teofilatto, Sep 30 2004

Except for the initial term, a(n)=={2, 4} mod 6.

Not monotonic: a(29) = 214 > 212 = a(30), a(33) = 272 > 268 = a(34), etc. - Charles R Greathouse IV, Jun 03 2013

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A001223, A001748, A047235, A100021.

ListConvolve[{-1,3},Prime[Range[100]]] (* Paolo Xausa, Nov 02 2023 *)

a(n) = 3*prime(n) - prime(n+1) \\ Michel Marcus, Jun 03 2013

a(n) = A001043(n) - 2*A001223(n).
a(n) = 3*A000040(n)-A000040(n+1) = A001748(n)-A000040(n+1) = A001747(n+1)-A001223(n). - R. J. Mathar, Apr 22 2010
a(n) ~ 2n log n. - Charles R Greathouse IV, Jun 03 2013
a(n) = A100021(n) + 3. - Hugo Pfoertner, Nov 02 2023
a(n) = A062234(n) + A000040(n). - Anthony S. Wright, Feb 19 2024

Corrected (116 duplicated) by R. J. Mathar, Apr 22 2010

A235700 a(n+1) = a(n) + (a(n) mod 5), a(1)=1.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 14, 18, 21, 22, 24, 28, 31, 32, 34, 38, 41, 42, 44, 48, 51, 52, 54, 58, 61, 62, 64, 68, 71, 72, 74, 78, 81, 82, 84, 88, 91, 92, 94, 98, 101, 102, 104, 108, 111, 112, 114, 118, 121, 122, 124, 128, 131, 132, 134, 138, 141, 142, 144, 148, 151, 152, 154, 158, 161, 162, 164, 168, 171, 172, 174, 178, 181, 182, 184, 188, 191

Offset: 1

M. F. Hasler, Jan 14 2014

Although the present sequence has not been thought of via "writing a(n) in base b", this could be seen as "base 5" version of A102039 (base 10) and A001651 (base 3), A047235 (base 6), A047350 (base 7) and A007612 (base 9). For 4 or 8 one would get a sequence constant from that (3rd resp. 4th) term on.

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

Cf. A001651, A007612, A047235, A047350, A102039.

NestList[#+Mod[#,5]&,1,80] (* Harvey P. Dale, Oct 20 2024 *)

is_A235700(n) = bittest(278,n%10) \\ 278=2^1+2^2+2^4+2^8

PARI
```
A235700 = n -> 2^((n-1)%4)+(n-1)\4*10
```

print1(a=1);for(i=1,99,print1(","a+=a%5))

Vec(x*(2*x^3+2*x^2+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Colin Barker, Jan 16 2014

a(n) = 2^(n-1 mod 4) + 10*floor((n-1)/4).
From Colin Barker, Jan 16 2014: (Start)
a(n) = (-10+(1+2*i)*(-i)^n+(1-2*i)*i^n+10*n)/4 where i=sqrt(-1).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4).
G.f.: x*(2*x^3+2*x^2+1) / ((x-1)^2*(x^2+1)). (End)
E.g.f.: (4 + 5*exp(x)*(x - 1) + cos(x) + 2*sin(x))/2. - Stefano Spezia, Feb 22 2025

A096689 Numbers n such that 2n^2 + 3n + 3 is prime.

Original entry on oeis.org

0, 2, 4, 10, 16, 20, 26, 34, 40, 44, 46, 50, 62, 64, 74, 76, 80, 82, 86, 92, 94, 110, 122, 140, 160, 164, 170, 176, 182, 200, 202, 212, 214, 220, 224, 232, 236, 250, 262, 296, 302, 304, 310, 320, 322, 326, 332, 344, 346, 352, 392, 400, 404, 422, 424, 446, 452

Offset: 1

Ray Chandler, Jul 12 2004

nonn

All n are {2,4} (mod 6), as in A047235, because otherwise 2*n^2 + 3*n + 3 is a multiple of 2 or 3. - R. J. Mathar, Jul 17 2012

Cf. A095697, A096690, A096691.

f[n_]:=n^2+(n+1)^2+(n+2); lst={};Do[p=f[n];If[PrimeQ[p],AppendTo[lst,n]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 04 2009 *)
Select[Range[0,500],PrimeQ[2#^2+3#+3]&] (* Harvey P. Dale, Jul 19 2011 *)

is(n)=isprime(2*n^2+3*n+3) \\ Charles R Greathouse IV, May 22 2017

a(n) = A096691(n)*2.

A104160 Primes equal to a sum of primes with differences congruent to (2,4) mod 6.

Original entry on oeis.org

353, 41, 131, 131, 311, 1181, 941, 1049, 1931, 2579, 3911, 4289, 4451, 6719, 8069, 10391, 10589, 12011, 14369, 26591, 31379, 33521, 35339, 41081, 43889, 58271, 59981, 63059, 64679, 66821, 115331, 74759, 77999, 78791, 80051, 80141, 83219, 87071, 94541, 96179

Offset: 1

Giovanni Teofilatto, Mar 10 2005

Consider finite ordered subsequences of at least 2 distinct primes A000040 subject to the conditions:
(i) the first differences of the subsequence are the initial terms of A047235,
(ii) the sum of the terms of the subsequence is a prime,
(iii) the subsequence is maximum in the sense that it cannot be extended by appending larger primes and still maintaining the conditions (i) and (ii).
Then the (prime) sum of the subsequence is one term of this sequence here.
The terms are inserted in order of the smallest prime in the subsequence.

			a(1)=353 because 353 = 5+7+11+19+29+43+59+79+101.
a(2)=41 because 41 = 11+13+17.
a(3)=131 because 131 = 17+19+23+31+41.
a(4)=131 because 131 = 41+43+47.
a(5)=311 because 311 = 101+103+107.

41 inserted, 131 duplicated, 311 inserted and sequence extended and comment added by R. J. Mathar, Apr 23 2010

A116127 Number of numbers that are congruent to {2, 4} mod 6 between prime(n) and prime(n+1) inclusive.

Original entry on oeis.org

1, 1, 0, 2, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 4, 2, 2, 0, 4, 0, 2, 2, 2, 2, 2, 0, 4, 0, 2, 0, 4, 4, 2, 0, 2, 2, 0, 4, 2, 2, 2, 0, 2, 2, 0, 4, 4, 2, 0, 2, 4, 2, 4, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 0, 4, 0, 2, 2, 2, 2, 2, 0, 2, 4, 2, 2, 2, 2, 2, 4, 0, 6, 2, 4, 2, 2, 0, 2

Offset: 1

Giovanni Teofilatto, Apr 08 2007

For n > 2,
A001223(n) = 2 iff a(n) = 0,
A001223(n) = 4 or 6 or 8 iff a(n) = 2,
A001223(n) = 10 or 12 or 14 iff a(n) = 4,
A001223(n) = 16 or 18 or 20 iff a(n) = 6,
and so on. This can be generalized to
A001223(n) = 3*k-2 or 3*k or 3*k+2 iff a(n) = k for k >= 2.

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

Cf. A000040 (primes), A001223 (differences between consecutive primes), A047235 (numbers congruent to {2, 4} mod 6), A002654.

[ #[ k: k in [NthPrime(n)..NthPrime(n+1)] | r eq 2 or r eq 4 where r is k mod 6 ]: n in [1..105] ]; /* Klaus Brockhaus, Apr 15 2007 */

P:= select(isprime, [seq(i,i=5..1000,2)]):
Delta:= P[2..-1]-P[1..-2]:
f:= t -> (t + 2*(t+1 mod 3) - 2)/3:
1,1,op(map(f, Delta)); # Robert Israel, Jun 19 2019

s={};Do[c=0;Do[If[MemberQ[{2,4},Mod[i,6]],c=c+1],{i,Prime[n],Prime[n+1]}];AppendTo[s,c],{n,105}];s (* James C. McMahon, Aug 18 2024 *)

Edited, corrected and extended by Klaus Brockhaus, Apr 15 2007

A123919 Number of numbers congruent to 2 or 4 mod 6 and <= n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 26

Offset: 1

Giovanni Teofilatto, Oct 29 2006

First differences of A056827. - R. J. Mathar, Nov 22 2008

a(n+2) is the graph radius of the n X n knight graph for n > 7. - Eric W. Weisstein, Nov 20 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Graph Radius
Eric Weisstein's World of Mathematics, Knight Graph
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A047235, A056827, A120325, A004526, A152467.

a:=[0,1,1,2,2,2,2];; for n in [8..80] do a[n]:=a[n-1]+a[n-6]-a[n-7]; od; a; # G. C. Greubel, Aug 07 2019

[Floor(n/2) - Floor(n/6) : n in [1..100]]; // Wesley Ivan Hurt, Apr 26 2021

a[n_] := Floor[n/2] - Floor[n/6]; Array[a, 80] (* Robert G. Wilson v, Oct 29 2006 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 1, 2, 2, 2, 2}, 80] (* G. C. Greubel, Aug 07 2019 *)

my(x='x+O('x^80)); concat([0], Vec(x^2*(1+x^2)/((1-x)*(1-x^6)))) \\ G. C. Greubel, Aug 07 2019

a(n) = floor(n/2) - floor(n/6);  \\ Joerg Arndt, Nov 23 2019

a(n) = floor(n/2) - floor(n/6).
From R. J. Mathar, Nov 22 2008: (Start)
G.f.: x^2*(1+x^2)/((1+x)*(1-x)^2*(1+x+x^2)*(1-x+x^2)).
a(n+1) - a(n) = A120325(n+1). (End)
a(n) = A004526(n) - A152467(n). - Omar E. Pol, Nov 25 2019
a(n) = a(n-1)+a(n-6)-a(n-7). - Wesley Ivan Hurt, Apr 26 2021
a(n) = floor((2*n+3+(-1)^n)/6). - Adriano Caroli, Mar 14 2025

A131717 Natural numbers A000027 with 6n+4 and 6n+5 terms swapped.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 23, 22, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 35, 34, 36, 37, 38, 39, 41, 40, 42, 43, 44, 45, 47, 46, 48, 49, 50, 51, 53, 52, 54, 55, 56, 57, 59, 58, 60, 61, 62, 63, 65, 64, 66, 67, 68, 69, 71, 70, 72

Offset: 1

Paul Curtz, Sep 15 2007

Hexaperiodic differences: 1, 1, 2, -1, 2, 1; 0, 1, -3, 3, -1, 0 (even palindromic signed); 1,-4, 6, -4, 1, 0.

Frieder Mittmann, Table of n, a(n) for n = 1..1002
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A131042.

seq(seq(6*i+s,s=[1,2,3,5,4,6]),i=0..100); # Robert Israel, Nov 11 2014

Drop[CoefficientList[Series[x (2x^5 - x^4 + 2x^3 + x^2 + x + 1)/((x - 1)^2 (x + 1) (x^2 - x + 1) (x^2 + x + 1)), {x, 0, 100}], x], 1] (* Indranil Ghosh, Apr 18 2017 *)
Table[Sum[(7 #1 - 13 #2 + 17 #3 - 3 #4 + 2 #5 + 2 #6)/30 & @@ Mod[k + Range[0, 5], 6], {k, 0, n}], {n, 0, 71}] (* Michael De Vlieger, Apr 22 2017 *)

Vec(x*(2*x^5-x^4+2*x^3+x^2+x+1)/((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 11 2014

a(n) = A008585(n/3) if n is congruent to 0 mod 3. - Frieder Mittmann, Nov 11 2014
a(n) = A007310((n-1)/3) if n is congruent to 1 mod 3. - Frieder Mittmann, Nov 11 2014
a(n) = A047235((n-2)/3) if n is congruent to 2 mod 3. - Frieder Mittmann, Nov 11 2014
G.f.: x*(2*x^5-x^4+2*x^3+x^2+x+1) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Nov 11 2014
a(n) = (24*floor(n/6)-3*(n^2-3*n-2)-9*floor(n/3)*(3*floor(n/3)-2*n+3)+(-1)^floor(n/3)*(3*n^2-5*n-6+3*floor(n/3)*(9*floor(n/3)-6*n+5)))/4. - Luce ETIENNE, Apr 18 2017

A154264 Nonnegative numbers n such that 9n^2 - 10n + 3 is prime.

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 22, 26, 28, 34, 38, 44, 46, 50, 52, 68, 70, 82, 86, 88, 100, 112, 118, 122, 124, 140, 146, 148, 152, 158, 170, 182, 188, 190, 196, 212, 224, 232, 236, 248, 256, 278, 280, 284, 286, 290, 292, 298, 310, 314, 334, 356, 374, 376, 380, 388, 394, 400

Offset: 1

Vincenzo Librandi, Feb 21 2009

With the exception of the 1, all entries are == 2 (mod 6) or == 4 (mod 6), as in A047235, because otherwise 9*n^2-10*n+3 is divisible by 2 or 3. - R. J. Mathar, Jul 16 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A154261.

[n: n in [0..400] | IsPrime(9*n^2-10*n+3)]; // Vincenzo Librandi, Sep 24 2012

lst={};Do[p=9*n^2-10*n+3;If[PrimeQ[p],AppendTo[lst,n]],{n,0,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 21 2009 *)

is(n)=isprime(9*n^2-10*n+3) \\ Charles R Greathouse IV, May 22 2017

Edited by Robert Hochberg, Jun 21 2010

A203016 Numbers congruent to {1, 2, 3, 4} mod 6, multiplied by 3.

Original entry on oeis.org

3, 6, 9, 12, 21, 24, 27, 30, 39, 42, 45, 48, 57, 60, 63, 66, 75, 78, 81, 84, 93, 96, 99, 102, 111, 114, 117, 120, 129, 132, 135, 138, 147, 150, 153, 156, 165, 168, 171, 174, 183, 186, 189, 192, 201, 204, 207, 210, 219, 222, 225, 228, 237, 240, 243, 246, 255, 258, 261, 264, 273, 276, 279, 282, 291, 294, 297

Offset: 1

N. J. A. Sloane, Dec 27 2011

Appears to coincide with the list of numbers n such that A006600(n) is not a multiple of n. Equals A047227 multiplied by 3.

Colin Foster, Peripheral mathematical knowledge, For the Learning of Mathematics, vol. 31, #3 (November, 2011), pp. 24-28.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A006600, A047227, A047235, A047241.

[3*n : n in [0..100] | n mod 6 in [1..4]]; // Wesley Ivan Hurt, Jun 07 2016

A203016:=n->3*(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A203016(n), n=1..100); # Wesley Ivan Hurt, Jun 07 2016

3 Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]] &] (* Wesley Ivan Hurt, Jun 07 2016 *)

From Wesley Ivan Hurt, Jun 07 2016: (Start)
G.f.: 3*x*(1+x+x^2+x^3+2*x^4)/((x-1)^2*(1+x+x^2+x^3)).
a(n) = 3*(6*n-5-i^(2*n)+(1+i)*i^(1-n)+(1-i)*i^(n-1))/4 where i=sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = 3*A047235(k), a(2k-1) = 3*A047241(k). (End)
E.g.f.: 3*(4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 07 2016

cp's OEIS Frontend

A093719 a(n) = (n mod 2)^(n mod 3).

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A098764 a(n) = 3p - q where p and q are consecutive primes.

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A235700 a(n+1) = a(n) + (a(n) mod 5), a(1)=1.

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A096689 Numbers n such that 2n^2 + 3n + 3 is prime.

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A104160 Primes equal to a sum of primes with differences congruent to (2,4) mod 6.

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A116127 Number of numbers that are congruent to {2, 4} mod 6 between prime(n) and prime(n+1) inclusive.

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Crossrefs

Programs

Magma

Maple

Mathematica

Extensions

A123919 Number of numbers congruent to 2 or 4 mod 6 and <= n.

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Crossrefs

Programs

GAP

Magma

A154264 Nonnegative numbers n such that 9n^2 - 10n + 3 is prime.