A153238 -id:A153238 - cp's OEIS frontend

A153644 a(n) = 4n^2 + 28n + 10.

42, 82, 130, 186, 250, 322, 402, 490, 586, 690, 802, 922, 1050, 1186, 1330, 1482, 1642, 1810, 1986, 2170, 2362, 2562, 2770, 2986, 3210, 3442, 3682, 3930, 4186, 4450, 4722, 5002, 5290, 5586, 5890, 6202, 6522, 6850, 7186, 7530, 7882, 8242, 8610, 8986, 9370

Offset: 1

Vincenzo Librandi, Dec 30 2008

Sequence gives values of x such that x^3 + 39x^2 = y^2 since a(n)^3 + 39*a(n)^2 = (8n^3 + 84n^2 + 216n + 70)^2.
a(n) = 2*(seventh diagonal to A153238).
About the first comment, naturally, the complete list of nonnegative values of x in x^3 + 39*x^2 = y^2 is given by x = m^2-39 with m>6. - Bruno Berselli, Jan 25 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A153238, A067076.

[4*n^2 + 28*n + 10: n in [1..50]]; // Vincenzo Librandi, Jan 25 2012

LinearRecurrence[{3,-3,1},{42,82,130}, 25] (* G. C. Greubel, Aug 23 2016 *)
Table[4n^2+28n+10,{n,70}] (* Harvey P. Dale, Jan 15 2023 *)

a(n)=4*n*(n+7)+10 \\ Charles R Greathouse IV, Jan 24 2012

From Colin Barker, Jan 24 2012: (Start)
a(1)=42, a(2)=82, a(3)=130, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 2*x*((3-x)*(7-5*x))/(1-x)^3. (End)
E.g.f.: 2*(-5 + (5 + 16*x + 2*x^2)*exp(x)). - G. C. Greubel, Aug 23 2016
Sum_{n>=1} 1/a(n) = 62/1995 + tan(sqrt(39)*Pi/2)*Pi/(4*sqrt(39)). - Amiram Eldar, Mar 02 2023

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

A282194 a(n) = smallest positive k such that 2*n + 2^k + 1 is composite.

Original entry on oeis.org

3, 5, 2, 1, 4, 2, 1, 7, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 4, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 4, 2, 1, 4, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Altug Alkan, Feb 15 2017

nonn

Least k such that a(k) = n are 3, 2, 0, 4, 1, 112, 7, 32917, 802, 9712, 1198673602 for the initial terms.

			a(1) = 5 because 3 + 2^k is prime for 0 < k < 5 and 3 + 2^5 = 35 is composite.

Altug Alkan, Table of n, a(n) for n = 0..10000

Cf. A067076, A067760, A153238, A282026.

spk[n_]:=Module[{k=1},While[!CompositeQ[2n+2^k+1],k++];k]; Array[spk,110,0] (* Harvey P. Dale, Apr 26 2017 *)

a(n) = my(k=1); while(isprime(2*n+2^k+1), k++); k;

A308833 Numbers r such that the r-th tetrahedral number A000292(r) divides r!.

Original entry on oeis.org

1, 7, 8, 13, 14, 19, 20, 23, 24, 25, 26, 31, 32, 33, 34, 37, 38, 43, 44, 47, 48, 49, 50, 53, 54, 55, 56, 61, 62, 63, 64, 67, 68, 73, 74, 75, 76, 79, 80, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 97, 98, 103, 104, 109, 110, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Lorenzo Sauras Altuzarra, Jun 28 2019

Conjecture: for every odd integer r > 1, the following statements are equivalent: a) r is a term of this sequence, b) r + 1 is a term of this sequence, c) r + 2 is composite.

			The 7th tetrahedral number is 84, and 84*60 = 5040 = 7!.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000142 (factorial numbers), A000292 (tetrahedral numbers).

Cf. A007921 (numbers which are not difference of two primes), A153238.

q := n -> (irem(n!, n*(n+1)*(n+2)/6) = 0):
select(q, [$1..120])[];

Select[Range@ 120, Mod[#!, Pochhammer[#, 3]/6] == 0 &] (* Michael De Vlieger, Jul 08 2019 *)

isok(k) = !(k! % (k*(k+1)*(k+2)/6)); \\ Michel Marcus, Jun 28 2019

is(n) = { my(f = factor(binomial(n + 2, 3))); forstep(i = #f~, 1, -1, if(val(n, f[i, 1]) - f[i, 2] < 0, return(0) ) ); 1 }
val(n, p) = my(r=0); while(n, r+=n\=p);r \\ David A. Corneth, Mar 22 2021

A153261 Primes p such that 3*p-2 is not prime.

Original entry on oeis.org

2, 17, 19, 29, 31, 41, 59, 73, 79, 83, 89, 97, 101, 107, 109, 131, 139, 149, 151, 157, 173, 179, 197, 199, 223, 227, 229, 233, 239, 241, 269, 281, 283, 311, 317, 349, 353, 359, 367, 379, 383, 389, 397, 409, 419, 421, 439, 449, 457, 463, 479, 499, 503, 509, 521

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

nonn

3*2-2=4 not prime, 17*3-2=49 not prime,... 3*3-2=7 is prime and not in this sequence.

Cf. A153184, A088878

Cf. A153238 [From Vincenzo Librandi, Jan 01 2009]

lst={};Do[p=Prime[n];If[ !PrimeQ[3*p-2],AppendTo[lst,p]],{n,6!}];lst

Minor edits by N. J. A. Sloane, Jul 08 2010

cp's OEIS Frontend

A153644 a(n) = 4*n^2 + 28*n + 10.

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Magma

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

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Magma

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

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

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A282194 a(n) = smallest positive k such that 2*n + 2^k + 1 is composite.

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A308833 Numbers r such that the r-th tetrahedral number A000292(r) divides r!.

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Maple

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PARI

A153261 Primes p such that 3*p-2 is not prime.

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A153644 a(n) = 4n^2 + 28n + 10.

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

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

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