A144562 -id:A144562 - cp's OEIS frontend

A142555 Primes congruent to 25 mod 53.

131, 449, 661, 1297, 1721, 1933, 2039, 2251, 2357, 2887, 3947, 4159, 4583, 5113, 5431, 5749, 6067, 6173, 6491, 6703, 7127, 8081, 8293, 8929, 9883, 10837, 11261, 11579, 11897, 12109, 13063, 13381, 13487, 14653, 14759, 15077, 15289, 15607, 16349, 16561, 16879

Offset: 1

N. J. A. Sloane, Jul 11 2008

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

Cf. A000040, A144562.

[p: p in PrimesUpTo(20000) | p mod 53 eq 25]; // Vincenzo Librandi, Aug 30 2012

Select[Prime[Range[2600]], MemberQ[{25}, Mod[#, 53]] &] (* Vincenzo Librandi, Aug 30 2012 *)
Select[Range[25,17000,53],PrimeQ] (* Harvey P. Dale, Jul 27 2013 *)

is(n)=isprime(n) && n%53==25 \\ Charles R Greathouse IV, Jul 03 2016

a(n) ~ 52n log n. - Charles R Greathouse IV, Jul 03 2016

A142755 Primes congruent to 28 mod 59.

Original entry on oeis.org

677, 1031, 1621, 2447, 2683, 2801, 3037, 3391, 3863, 4099, 4217, 5279, 5869, 5987, 6577, 7639, 7757, 7993, 8111, 8819, 9173, 10589, 11887, 12241, 12713, 13421, 14011, 15073, 15427, 16253, 16607, 16843, 17551, 17669, 18731, 19793, 20029, 20147, 21563, 21799

Offset: 1

N. J. A. Sloane, Jul 11 2008

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

Cf. A000040, A144562.

[p: p in PrimesUpTo(22000) | p mod 59 eq 28 ]; // Vincenzo Librandi, Sep 03 2012

Select[59*Range[400]+28,PrimeQ] (* Harvey P. Dale, Jul 19 2011 *)
Select[Prime[Range[2600]], MemberQ[{28}, Mod[#, 59]] &] (* Vincenzo Librandi, Sep 03 2012 *)

is(n)=isprime(n) && n%59==28 \\ Charles R Greathouse IV, Jul 03 2016

a(n) ~ 58n log n. - Charles R Greathouse IV, Jul 03 2016

A142827 Primes congruent to 29 mod 61.

Original entry on oeis.org

29, 151, 761, 883, 1249, 1493, 2347, 2591, 2713, 2957, 3079, 3323, 4177, 4421, 4787, 4909, 5153, 5519, 5641, 6007, 6373, 6983, 7349, 8081, 8447, 10399, 11131, 11497, 11863, 12107, 12473, 13327, 13693, 14303, 14669, 15401, 15767, 15889, 16987, 17231, 17597

Offset: 1

N. J. A. Sloane, Jul 11 2008

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

Cf. A000040, A144562.

[p: p in PrimesUpTo(18000) | p mod 61 eq 29 ] ; // Vincenzo Librandi, Sep 05 2012

Select[Prime[Range[2100]],Mod[#,61]==29&]  (* Harvey P. Dale, Mar 30 2011 *)
Select[Range[29,18000,61],PrimeQ] (* Harvey P. Dale, Nov 19 2011 *) (* For a larger range of numbers, the second program is faster *)
Select[Prime[Range[2300]], MemberQ[{29}, Mod[#, 61]] &] (* Vincenzo Librandi, Sep 05 2012 *)

is(n)=isprime(n) && n%61==29 \\ Charles R Greathouse IV, Jul 03 2016

a(n) ~ 60n log n. - Charles R Greathouse IV, Jul 03 2016

A154576 a(n) = 2n^2 + 14n + 5.

Original entry on oeis.org

21, 41, 65, 93, 125, 161, 201, 245, 293, 345, 401, 461, 525, 593, 665, 741, 821, 905, 993, 1085, 1181, 1281, 1385, 1493, 1605, 1721, 1841, 1965, 2093, 2225, 2361, 2501, 2645, 2793, 2945, 3101, 3261, 3425, 3593, 3765, 3941, 4121, 4305, 4493, 4685, 4881

Offset: 1

Vincenzo Librandi, Jan 12 2009

Seventh diagonal in A144562.

2*a(n) + 39 is a square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A144562, A154577.

I:=[21, 41, 65]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 22 2012

LinearRecurrence[{3, -3, 1}, {21, 41, 65}, 50] (* Vincenzo Librandi, Feb 22 2012 *)

for(n=1, 40, print1(2*n^2 + 14*n + 5", ")); \\ Vincenzo Librandi, Feb 22 2012

G.f.: x*(3-x)*(7-5*x)/(1-x)^3. - Bruno Berselli, Dec 07 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 22 2012
Sum_{n>=1} 1/a(n) = 124/1995 + tan(sqrt(39)*Pi/2)*Pi/(2*sqrt(39)). - Amiram Eldar, Feb 25 2023

A154591 a(n) = 2n^2 + 18n + 7.

Original entry on oeis.org

27, 51, 79, 111, 147, 187, 231, 279, 331, 387, 447, 511, 579, 651, 727, 807, 891, 979, 1071, 1167, 1267, 1371, 1479, 1591, 1707, 1827, 1951, 2079, 2211, 2347, 2487, 2631, 2779, 2931, 3087, 3247, 3411, 3579, 3751, 3927, 4107, 4291, 4479, 4671, 4867, 5067, 5271

Offset: 1

Vincenzo Librandi, Jan 12 2009

Ninth diagonal of A144562.

2*a(n) + 67 is a square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067076, A144562, A153238.

I:=[27, 51, 79]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 22 2012

LinearRecurrence[{3, -3, 1}, {27, 51, 79}, 50] (* Vincenzo Librandi, Feb 22 2012 *)

for(n=1, 40, print1(2*n^2 + 18*n + 7", ")); \\ Vincenzo Librandi, Feb 22 2012

[2*n^2+18*n+7 for n in range(1,51)] #  G. C. Greubel, May 27 2024

G.f.: (9*x^2-6*x-7)/(x-1)^3. - Bruno Berselli, Dec 07 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 22 2012
Sum_{n>=1} 1/a(n) = 1621/20097 + tan(sqrt(67)*Pi/2)*Pi/(2*sqrt(67)). - Amiram Eldar, Feb 25 2023
E.g.f.: (7 + 20*x + 2*x^2)*exp(x). - G. C. Greubel, May 27 2024

A154600 a(n) = 2n^2 + 22n + 9.

Original entry on oeis.org

33, 61, 93, 129, 169, 213, 261, 313, 369, 429, 493, 561, 633, 709, 789, 873, 961, 1053, 1149, 1249, 1353, 1461, 1573, 1689, 1809, 1933, 2061, 2193, 2329, 2469, 2613, 2761, 2913, 3069, 3229, 3393, 3561, 3733, 3909, 4089, 4273, 4461, 4653, 4849, 5049, 5253

Offset: 1

Vincenzo Librandi, Jan 12 2009

Eleventh diagonal of A144562.

2*a(n) + 103 is a square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A144562.

I:=[33, 61, 93]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 26 2012

LinearRecurrence[{3, -3, 1}, {33, 61, 93}, 50] (* Vincenzo Librandi, Feb 26 2012 *)
4*Binomial[Range[50]+6,2] - 51 (* G. C. Greubel, May 30 2024 *)

a(n)=2*n*(n+22)+9 \\ Charles R Greathouse IV, Jan 11 2012

[2*n^2+22*n+9 for n in range(1,51)] # G. C. Greubel, May 30 2024

From Vincenzo Librandi, Feb 26 2012: (Start)
G.f: x*(33 - 38*x + 9*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
Sum_{n>=1} 1/a(n) = 257162/3084939 + tan(sqrt(103)*Pi/2)*Pi/(2*sqrt(103)). - Amiram Eldar, Feb 25 2023
E.g.f.: -9 + (9 + 24*x + 2*x^2)*exp(x). - G. C. Greubel, May 30 2024

A154628 Primes congruent to 35 mod 73.

Original entry on oeis.org

181, 619, 911, 1787, 1933, 2371, 2663, 3539, 4561, 4999, 5437, 7043, 7481, 7919, 8941, 9817, 11131, 11423, 12007, 13613, 13759, 14051, 14197, 14489, 15073, 15511, 15803, 17117, 18869, 19891, 20183, 21059, 22811, 23687, 23833, 24709, 25147

Offset: 1

Vincenzo Librandi, Jan 17 2009

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

Cf. A000040, A144562.

[p: p in PrimesUpTo(26000) | p mod 73 eq 35]; // Vincenzo Librandi, Sep 07 2012

Select[Prime[Range[3000]],Mod[#,73]==35&] (* Harvey P. Dale, May 21 2011 *)
Select[Prime[Range[3000]], MemberQ[{35}, Mod[#, 73]] &] (* Vincenzo Librandi, Sep 07 2012 *)

is(n)=isprime(n) && n%73==35 \\ Charles R Greathouse IV, Jul 03 2016

a(n) ~ 72n log n. - Charles R Greathouse IV, Jul 03 2016

A154575 a(n) = 2n^2 + 12n + 4.

Original entry on oeis.org

18, 36, 58, 84, 114, 148, 186, 228, 274, 324, 378, 436, 498, 564, 634, 708, 786, 868, 954, 1044, 1138, 1236, 1338, 1444, 1554, 1668, 1786, 1908, 2034, 2164, 2298, 2436, 2578, 2724, 2874, 3028, 3186, 3348, 3514, 3684, 3858, 4036, 4218, 4404, 4594, 4788, 4986, 5188

Offset: 1

Vincenzo Librandi, Jan 12 2009

Sixth diagonal of A144562.

2*a(n) + 28 is a square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028881, A067076, A144562, A153238.

I:=[18, 36, 58]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 26 2012

LinearRecurrence[{3, -3, 1}, {18, 36, 58}, 50] (* Vincenzo Librandi, Feb 26 2012 *)
Table[2n^2+12n+4,{n,50}] (* Harvey P. Dale, Sep 18 2019 *)

for(n=1, 50, print1(2*n^2+12*n+4", ")); \\ Vincenzo Librandi, Feb 26 2012

From R. J. Mathar, Jan 05 2011: (Start)
a(n) = 2*A028881(n+3).
G.f.: -2*x*(2*x-3)*(x-3)/(x-1)^3. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 26 2012
From Amiram Eldar, Feb 25 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/28 - cot(sqrt(7)*Pi)*Pi/(4*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 31/84 - cosec(sqrt(7)*Pi)*Pi/(4*sqrt(7)). (End)
E.g.f.: 2*exp(x)*(x^2 + 7*x + 2). - Elmo R. Oliveira, Nov 02 2024

A154590 a(n) = 2n^2 + 16n + 6.

Original entry on oeis.org

24, 46, 72, 102, 136, 174, 216, 262, 312, 366, 424, 486, 552, 622, 696, 774, 856, 942, 1032, 1126, 1224, 1326, 1432, 1542, 1656, 1774, 1896, 2022, 2152, 2286, 2424, 2566, 2712, 2862, 3016, 3174, 3336, 3502, 3672, 3846, 4024, 4206, 4392, 4582, 4776, 4974, 5176

Offset: 1

Vincenzo Librandi, Jan 12 2009

Eighth diagonal of A144562.

2*a(n) + 52 is a square.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067076, A116711, A144562, A153238.

Table[2n^2+16n+6,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{24,46,72},50] (* Harvey P. Dale, Dec 27 2011 *)

a(n)=2*n^2+16*n+6 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*A116711(n+3).
G.f.: -2*x*(3*x-4)*(x-3)/(x-1)^3.
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 35/468 - cot(sqrt(13)*Pi)*Pi/(4*sqrt(13)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 121/468 + cosec(sqrt(13)*Pi)*Pi/(4*sqrt(13)). (End)
From Elmo R. Oliveira, Jun 04 2025: (Start)
E.g.f.: 2*(exp(x)*(x^2 + 9*x + 3) - 3).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Corrected (a(31) added) by Harvey P. Dale, Dec 27 2011

A154599 a(n) = 2n^2 + 20n + 8.

Original entry on oeis.org

30, 56, 86, 120, 158, 200, 246, 296, 350, 408, 470, 536, 606, 680, 758, 840, 926, 1016, 1110, 1208, 1310, 1416, 1526, 1640, 1758, 1880, 2006, 2136, 2270, 2408, 2550, 2696, 2846, 3000, 3158, 3320, 3486, 3656, 3830, 4008, 4190, 4376, 4566, 4760, 4958, 5160

Offset: 1

Vincenzo Librandi, Jan 12 2009

Tenth diagonal of A144562.

2*a(n) + 84 is a square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067076, A127147, A144562, A153238.

I:=[30, 56, 86]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 26 2012

LinearRecurrence[{3, -3, 1}, {30, 56, 86}, 50] (* Vincenzo Librandi, Feb 26 2012 *)
Table[2n^2+20n+8,{n,50}] (* Harvey P. Dale, Jun 15 2019 *)

for(n=1, 40, print1(2*n^2+20*n+8", ")); \\ Vincenzo Librandi, Feb 26 2012

[2*n^2+20*n+8 for n in range(1,41)] # G. C. Greubel, May 30 2024

From R. J. Mathar, Jan 05 2011: (Start)
a(n) = 2*A127147(n+13).
G.f.: 2*x*(5-4*x)*(3-x)/(1-x)^3. (End)
From Amiram Eldar, Feb 25 2023: (Start)
Sum_{n>=1} 1/a(n) = 79/952 - cot(sqrt(21)*Pi)*Pi/(4*sqrt(21)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2851/14280 - cosec(sqrt(21)*Pi)*Pi/(4*sqrt(21)). (End)
E.g.f.: 2*(-4 + (4 + 11*x + x^2)*exp(x)). - G. C. Greubel, May 30 2024

cp's OEIS Frontend

A142555 Primes congruent to 25 mod 53.

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A142755 Primes congruent to 28 mod 59.

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A142827 Primes congruent to 29 mod 61.

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A154576 a(n) = 2*n^2 + 14*n + 5.

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A154591 a(n) = 2*n^2 + 18*n + 7.

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A154600 a(n) = 2*n^2 + 22*n + 9.

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A154628 Primes congruent to 35 mod 73.

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Magma

A154576 a(n) = 2n^2 + 14n + 5.

A154591 a(n) = 2n^2 + 18n + 7.

A154600 a(n) = 2n^2 + 22n + 9.

A154575 a(n) = 2n^2 + 12n + 4.

A154590 a(n) = 2n^2 + 16n + 6.

A154599 a(n) = 2n^2 + 20n + 8.