A067076 -id:A067076 - 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

A160973 a(n) is the number of positive integers of the form (n-3k)/(2k+1), 1 <= k <= (n-1)/5.

Original entry on oeis.org

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

Offset: 0

Vladimir Shevelev, Jun 01 2009, Jun 07 2009

nonn

If n is different from 3, then a(n)=0 iff n is in A067076, i.e., 2n+3 is prime.

Cf. A067076, A034953, A054269, A082749, A006254, A161116.

a[n_] := Length[Select[Range[Floor[(n-1)/5]], IntegerQ[(n-3#)/(2#+1)] &]]; Array[a, 100, 0] (* Amiram Eldar, Dec 15 2018 *)

a(n) = sum(k=1, (n-1)/5, frac((n-3*k)/(2*k+1)) == 0); \\ Michel Marcus, Dec 15 2018

Edited by N. J. A. Sloane, Jun 07 2009

a(44) corrected and more terms from Michel Marcus, Dec 15 2018

A161116 a(n) is the number of nontrivial positive divisors of 2n+3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 2, 0, 0, 4, 0, 1, 2, 0, 2, 2, 0, 0, 4, 2, 0, 2, 0, 0, 4, 2, 0, 3, 0, 2, 2, 0, 2, 2, 2, 0, 4, 0, 0, 6, 0, 0, 2, 0, 2, 4, 2, 1, 2, 2, 0, 2, 0, 2, 6, 0, 0, 2, 2, 2, 4, 0, 0, 4, 2, 0, 2, 2, 0, 6, 0, 1, 4, 0, 4, 2, 0, 0, 2

Offset: 0

Vladimir Shevelev, Jun 02 2009

a(n)=0 iff n is in A067076, i.e., 2n+3 is prime; a(n) is the number of positive integers of the form (n-3k)/(2k+1), 1<=k<=n/3.

			Since for n=3 we have 2n+3=9 and only nontrivial divisor of 9 is 3, then a(3)=1.

Cf. A067076, A034953, A054269, A082749, A006254.

Cf. A160973. - Vladimir Shevelev, Jun 07 2009

[NumberOfDivisors(2*n+3)-2: n in [0..100]]; // Vincenzo Librandi, Feb 08 2016

a(n) = numdiv(2*n+3) - 2; \\ Michel Marcus, Feb 08 2016

For n>=1, a(n)=A160973(n)+A079978(n). [Vladimir Shevelev, Jun 07 2009]

a(n) = A070824(2n+3).

Edited by Charles R Greathouse IV, Oct 12 2009

More terms from Michel Marcus, Feb 08 2016

A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

Original entry on oeis.org

2, 5, 8, 14, 17, 20, 29, 32, 35, 38, 47, 50, 53, 62, 68, 74, 77, 80, 89, 95, 98, 104, 110, 113, 119, 134, 137, 140, 152, 155, 164, 167, 173, 182, 185, 188, 197, 203, 209, 215, 218, 227, 230, 242, 248, 260, 269, 272, 284, 287, 299

Offset: 1

Eric Desbiaux, Feb 11 2010

With Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR
second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 2 + 3 OR 3* 1 + 4 =  7;
  2* 5 + 3 OR 3* 3 + 4 = 13;
  2* 8 + 3 OR 3* 5 + 4 = 19;
  2*14 + 3 OR 3* 9 + 4 = 31;
  2*17 + 3 OR 3*11 + 4 = 37;
  2*20 + 3 OR 3*13 + 4 = 43;
  2*29 + 3 OR 3*19 + 4 = 61;
  2*32 + 3 OR 3*21 + 4 = 67;
  2*35 + 3 OR 3*23 + 4 = 73.
A034936 Numbers k such that 3k+4 is prime.
A002476 Primes of the form 6k+1.
A024899 Nonnegative integers k such that 6k+1 is prime.
2, 5, 8, 14, 17, 20, ... = (3*(4*A024899 - A034936) - 5)/2.

Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?

Cf. A067076, A034936, A002476, A024899.

Select[Range[300],PrimeQ[2#+3]&&Divisible[2#-1,3]&] (* Harvey P. Dale, Aug 25 2016 *)

More terms from Harvey P. Dale, Aug 25 2016

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;

A294064 Numbers k such that 2k - 3, 2k + 3, 3k - 2, 3k + 2 are primes.

Original entry on oeis.org

5, 7, 13, 35, 43, 55, 77, 127, 133, 155, 167, 253, 287, 295, 365, 475, 497, 533, 595, 713, 1007, 1177, 1483, 1805, 2323, 2575, 2723, 2927, 3107, 3415, 3487, 3823, 4145, 4213, 4367, 4565, 4717, 4927, 4963, 5125, 5215, 5363, 5417, 5587, 5627, 5795, 6133, 6587, 6797

Offset: 1

Dimitris Valianatos, Oct 22 2017

nonn

The common numbers of A098090, A067076, A153183, A024893.
Conjecture: The Sum_{n>=1} 1/a(n) = 0.57... converges.
Note that the sum of the 4 primes that are obtained is 10 times the original term: (2*k - 3) + (2*k + 3) + (3*k - 2) + (3*k + 2) = 10*k.
From Robert G. Wilson v, Nov 19 2017: (Start)
Number of terms less than 10^m: 2, 7, 20, 55, 189, 919, 4863, 28218, 174469, ..., ;
Number of prime terms less than 10^m: 2, 4, 6, 12, 39, 140, 558, 2755, 14804, ..., .
All terms are == {5, 7, 13, 17, 23, 25} (mod 30).
(End)

			5 is in the sequence because 2*5-3 = 7, 2*5+3 = 13, 3*5-2 = 13, 3*5+2 = 17 and the tetrad [7, 13, 13, 17] are all prime numbers.
7 is in the sequence because 2*7-3 = 11, 2*7+3 = 17, 3*7-2 = 19, 3*7+2 = 23 and the tetrad [11, 17, 19, 23] are all prime numbers.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A098090, A067076, A153183, A024893.

Select[Range[10^4], Function[k, AllTrue[Flatten@ Map[#1 k + {-1, 1} #2 & @@ # &, {#, Reverse@ #}] &@ {2, 3}, PrimeQ]]] (* Michael De Vlieger, Oct 22 2017 *)

{
for(n=1,10000,
    if(isprime(2*n-3)&&isprime(2*n+3)&&isprime(3*n-2)&&isprime(3*n+2),
       print1(n", ")
      )
   )
}

A337491 Numbers k such that exactly one of 2k + 3 and 4k + 3 is prime.

Original entry on oeis.org

8, 11, 13, 16, 22, 26, 28, 29, 31, 35, 37, 38, 41, 43, 44, 50, 53, 56, 59, 64, 65, 68, 70, 73, 74, 76, 80, 85, 86, 88, 91, 97, 98, 107, 109, 112, 113, 116, 118, 121, 122, 125, 127, 133, 134, 136, 137, 139, 142, 145, 146, 149, 151, 152, 155, 160, 161, 167, 170

Offset: 1

K. D. Bajpai, Aug 29 2020

nonn

Integers that are in A067076 or in A095278, but not in both. - Michel Marcus, Aug 29 2020

			a(1) = 8 is a term because 2*8 + 3 = 19 is a prime; but 4*8 + 3 = 35 = (5*7) is a composite number.
a(4) = 16 is a term because 2*16 + 3 = 35 = (5*7) is a composite number; but 4*16 + 3 = 67  is a prime.
a(6) = 26 is a term because 2*26 + 3 = 55 = (5*11) is a composite number; but 4*26 + 3 = 107  is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5000

Cf. A063908, A067076, A095278, A337480.

A337491:=n->`if`((isprime(2*n+3) xor isprime(4*n+3)), n, NULL): seq(A337491(n), n=1..500);

Select[Range[0, 250], Xor[PrimeQ[2 # + 3], PrimeQ[4 # + 3]] &]
Select[Range[200],Total[Boole[PrimeQ[{2,4}#+3]]]==1&] (* Harvey P. Dale, Jan 26 2023 *)

isok(k) = bitxor(isprime(2*k+3), isprime(4*k+3)); \\ Michel Marcus, Aug 29 2020

cp's OEIS Frontend

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

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

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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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A160973 a(n) is the number of positive integers of the form (n-3k)/(2k+1), 1 <= k <= (n-1)/5.

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A161116 a(n) is the number of nontrivial positive divisors of 2n+3.

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A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

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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.

A294064 Numbers k such that 2k - 3, 2k + 3, 3k - 2, 3k + 2 are primes.

A337491 Numbers k such that exactly one of 2k + 3 and 4k + 3 is prime.