A047348 -id:A047348 - cp's OEIS frontend

A002641 Numbers k such that (k^2 + k + 1)/7 is prime.

4, 9, 11, 23, 32, 39, 44, 51, 53, 60, 65, 72, 86, 93, 95, 114, 123, 156, 170, 179, 186, 200, 207, 212, 219, 228, 233, 240, 249, 261, 270, 303, 317, 333, 338, 345, 375, 389, 401, 443, 452, 473, 480, 492, 515, 534, 548, 564, 578, 585, 597, 599, 611, 641, 660, 662

Offset: 1

N. J. A. Sloane

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
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).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

Subsequence of A047348.

/* See Crossrefs: */
A047348:=[m: m in [1..700] | m mod 7 in [2,4]];
[n: n in A047348 | IsPrime( (n^2+n+1) div 7 )];
// Bruno Berselli, Sep 26 2012

Select[Range[700], PrimeQ[(#^2 + # + 1)/7] &] (* Harvey P. Dale, Sep 16 2011 *)

is(n)=isprime((n^2+n+1)/7) \\ Charles R Greathouse IV, May 22 2017

More terms from Jon E. Schoenfield, Mar 24 2010

A047300 Numbers that are congruent to {2, 3, 4, 6} mod 7.

Original entry on oeis.org

2, 3, 4, 6, 9, 10, 11, 13, 16, 17, 18, 20, 23, 24, 25, 27, 30, 31, 32, 34, 37, 38, 39, 41, 44, 45, 46, 48, 51, 52, 53, 55, 58, 59, 60, 62, 65, 66, 67, 69, 72, 73, 74, 76, 79, 80, 81, 83, 86, 87, 88, 90, 93, 94, 95, 97, 100, 101, 102, 104, 107, 108, 109, 111

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047280, A047348.

[n : n in [0..150] | n mod 7 in [2, 3, 4, 6]]; // Wesley Ivan Hurt, Jun 02 2016

A047300:=n->(14*n-5-I^(2*n)-(1-3*I)*I^(-n)-(1+3*I)*I^n)/8: seq(A047300(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Table[(14n-5-I^(2n)-(1-3*I)*I^(-n)-(1+3*I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 02 2016 *)

G.f.: x*(2+x+x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-5-i^(2*n)-(1-3*i)*i^(-n)-(1+3*i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047280(k), a(2k-1) = A047348(k). (End)

A047338 Numbers that are congruent to {1, 2, 3, 4} mod 7.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18, 22, 23, 24, 25, 29, 30, 31, 32, 36, 37, 38, 39, 43, 44, 45, 46, 50, 51, 52, 53, 57, 58, 59, 60, 64, 65, 66, 67, 71, 72, 73, 74, 78, 79, 80, 81, 85, 86, 87, 88, 92, 93, 94, 95, 99, 100, 101, 102, 106, 107, 108, 109

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047348, A047356.

[n : n in [0..150] | n mod 7 in [1, 2, 3, 4]]; // Wesley Ivan Hurt, May 23 2016

A047338:=n->(14*n-15-3*I^(2*n)-(3-3*I)*I^(-n)-(3+3*I)*I^n)/8: seq(A047338(n), n=1..100); # Wesley Ivan Hurt, May 23 2016

Table[(14n-15-3*I^(2n)-(3-3*I)*I^(-n)-(3+3*I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 23 2016 *)

G.f.: x*(1+x+x^2+x^3+3*x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, May 23 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14n-15-3*i^(2n)-(3-3*i)*i^(-n)-(3+3*i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047348(n), a(2n-1) = A047356(n). (End)
E.g.f.: (12 + 3*(sin(x) - cos(x)) + (7*x - 6)*sinh(x) + (7*x - 9)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

More terms from Wesley Ivan Hurt, May 23 2016

A047340 Numbers that are congruent to {0, 2, 3, 4} mod 7.

Original entry on oeis.org

0, 2, 3, 4, 7, 9, 10, 11, 14, 16, 17, 18, 21, 23, 24, 25, 28, 30, 31, 32, 35, 37, 38, 39, 42, 44, 45, 46, 49, 51, 52, 53, 56, 58, 59, 60, 63, 65, 66, 67, 70, 72, 73, 74, 77, 79, 80, 81, 84, 86, 87, 88, 91, 93, 94, 95, 98, 100, 101, 102, 105, 107, 108, 109, 112

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047348, A047355.

[n : n in [0..150] | n mod 7 in [0,2,3,4]]; // Vincenzo Librandi, Feb 17 2014

A047340:=n->(14*n-17-I^(2*n)-(3-I)*I^(-n)-(3+I)*I^n)/8: seq(A047340(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Select[Range[0,100],MemberQ[{0,2,3,4},Mod[#,7]]&] (* or *) LinearRecurrence[ {1,0,0,1,-1},{0,2,3,4,7},100] (* Harvey P. Dale, Feb 16 2014 *)
CoefficientList[Series[x (2 + x + x^2 + 3 x^3)/((1 + x) (1 + x^2) (x - 1)^2), {x, 0, 200}], x] (* Vincenzo Librandi, Feb 17 2014 *)

G.f.: x^2*(2+x+x^2+3*x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (14n-17-i^(2n)-(3-i)*i^(-n)-(3+i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047348(n), a(2n-1) = A047355(n). (End)

More terms from Vincenzo Librandi, Feb 17 2014

A047362 Numbers that are congruent to {2, 3, 4, 5} mod 7.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 11, 12, 16, 17, 18, 19, 23, 24, 25, 26, 30, 31, 32, 33, 37, 38, 39, 40, 44, 45, 46, 47, 51, 52, 53, 54, 58, 59, 60, 61, 65, 66, 67, 68, 72, 73, 74, 75, 79, 80, 81, 82, 86, 87, 88, 89, 93, 94, 95, 96, 100, 101, 102, 103, 107, 108, 109, 110

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047348, A047389.

[n : n in [0..150] | n mod 7 in [2, 3, 4, 5]]; // Wesley Ivan Hurt, Jun 03 2016

A047362:=n->(14*n-7-3*(I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n))/8: seq(A047362(n), n=1..100); # Wesley Ivan Hurt, Jun 03 2016

Select[Range[100], MemberQ[{2,3,4,5}, Mod[#,7]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {2,3,4,5,9}, 60] (* Harvey P. Dale, Oct 03 2015 *)

G.f.: x*(2*x^2+3*x+2)*(x^2-x+1) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, Jun 03 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-7-3*(i^(2*n)+(1-i)*i^(-n)+(1+i)*i^n))/8 where i=sqrt(-1).
a(2k) = A047389(k), a(2k-1) = A047348(k). (End)

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A002641 Numbers k such that (k^2 + k + 1)/7 is prime.

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A047300 Numbers that are congruent to {2, 3, 4, 6} mod 7.

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A047338 Numbers that are congruent to {1, 2, 3, 4} mod 7.

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A047340 Numbers that are congruent to {0, 2, 3, 4} mod 7.

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A047362 Numbers that are congruent to {2, 3, 4, 5} mod 7.

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