A016885 -id:A016885 - cp's OEIS frontend

A321483 a(n) = 7*2^n + (-1)^n.

8, 13, 29, 55, 113, 223, 449, 895, 1793, 3583, 7169, 14335, 28673, 57343, 114689, 229375, 458753, 917503, 1835009, 3670015, 7340033, 14680063, 29360129, 58720255, 117440513, 234881023, 469762049, 939524095, 1879048193, 3758096383, 7516192769, 15032385535

Offset: 0

Paul Curtz, Nov 11 2018

Difference table:
   8, 13, 29, 55, 113, 223, 449, ...
   5, 16, 26, 58, 110, 226, 446, 898, ...
  11, 10, 32, 52, 116, 220, 452, 892, 1796, ...
  -1, 22, 20, 64, 104, 232, 440, 904, 1784, 3592, ...
      -2, 44, 40, 128, 208, 464, 880, 1808, 3568, 7184, ...
etc.
Every diagonal is a sequence of the form k*2^m.
a(n) is divisible by
.  5 if n is a term of A004767,
. 11 if n is a term of A016885,
. 13 if n is a term of A017533.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000079, A001045, A005029, A010710, A014551, A062092, A062510, A070366, A175805, A199207, A206372.
Cf. A033999, A005009.
Subsequence of A001651, A160543, A160544.

a[n_] := 7*2^n + (-1)^n ; Array[a, 32, 0] (* Amiram Eldar, Nov 12 2018 *)
CoefficientList[Series[E^-x + 7 E^(2 x), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 12 2018 *)
LinearRecurrence[{1,2},{8,13},40] (* Harvey P. Dale, Mar 18 2022 *)

Vec((8 + 5*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 11 2018

O.g.f.: (8 + 5*x) / ((1 + x)*(1 - 2*x)). - Colin Barker, Nov 11 2018
E.g.f.: exp(-x) + 7*exp(2*x). - Stefano Spezia, Nov 12 2018
a(n) = a(n-1) + 2*a(n-2).
a(n) = 2*a(n-1) + 3*(-1)^n for n>0, a(0)=8.
a(2*k) = 7*4^k + 1, a(2*k+1) = 14*4^k - 1.
a(n) = A014551(n) + A014551(n-1) + A014551(n-2).
a(n) = 2^(n+3) - 3*A001045(n).
a(n) mod 9 = A070366(n+3).
a(n) + a(n+1) = 21*2^n.

Two terms corrected, and more terms added by Colin Barker, Nov 11 2018

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

Original entry on oeis.org

1, 5, 2, 13, 8, 3, 25, 18, 11, 4, 41, 32, 23, 14, 5, 61, 50, 39, 28, 17, 6, 85, 72, 59, 46, 33, 20, 7, 113, 98, 83, 68, 53, 38, 23, 8, 145, 128, 111, 94, 77, 60, 43, 26, 9, 181, 162, 143, 124, 105, 86, 67, 48, 29, 10, 221, 200, 179, 158, 137, 116, 95, 74, 53, 32, 11

Offset: 1

Stefano Spezia, Dec 20 2019

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).

			n\k|   0   1   2   3   4   5
---+------------------------
1  |   1
2  |   5   2
3  |  13   8   3
4  |  25  18  11   4
5  |  41  32  23  14   5
6  |  61  50  39  28  17   6
...
For n = 3 the matrix M is
      1, 2, 3
      2, 4, 6
      3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A005408, A330287.

Cf. A000027: diagonal; A001105: 2nd column; A001844: 1st column; A016789: 1st subdiagonal; A016885: 2nd subdiagonal; A017029: 3rd subdiagonal; A017221: 4th subdiagonal; A017461: 5th subdiagonal; A081436: row sums; A132209: 3rd column; A164284: 7th subdiagonal; A269044: 6th subdiagonal.

Flatten[Table[1+k-2n-2k*n+2n^2,{n,1,11},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]

O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]

A013856 a(n) = 10^(5*n + 3).

Original entry on oeis.org

1000, 100000000, 10000000000000, 1000000000000000000, 100000000000000000000000, 10000000000000000000000000000, 1000000000000000000000000000000000, 100000000000000000000000000000000000000, 10000000000000000000000000000000000000000000, 1000000000000000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (100000).

Subsequence of A011557.

Cf. A016885.

[10^(5*n+3): n in [0..10]]; // Vincenzo Librandi, Jul 08 2011

10^(5*Range[0, 10] + 3) (* Paolo Xausa, Mar 04 2025 *)

makelist(10^(5*n+3),n,0,20); /* Martin Ettl, Oct 21 2012 */

From Elmo R. Oliveira, Mar 02 2025: (Start)
G.f.: 1000/(1 - 100000*x).
E.g.f.: 1000*exp(100000*x).
a(n) = A011557(A016885(n)). (End)

A013896 a(n) = 20^(5*n + 3).

Original entry on oeis.org

8000, 25600000000, 81920000000000000, 262144000000000000000000, 838860800000000000000000000000, 2684354560000000000000000000000000000, 8589934592000000000000000000000000000000000, 27487790694400000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (3200000).

Subsequence of A009964.

Cf. A013824, A013856, A016885.

[20^(5*n+3): n in [0..10]]; // Vincenzo Librandi, May 27 2011

20^(5Range[0,20]+3) (* or *) NestList[3200000#&,8000,20] (* Harvey P. Dale, Dec 05 2021 *)

a(n) = 3200000*a(n-1), a(0)=8000. - Vincenzo Librandi, May 27 2011
From Elmo R. Oliveira, Jul 11 2025: (Start)
G.f.: 8000/(1-3200000*x).
E.g.f.: 8000*exp(3200000*x).
a(n) = A013824(n)*A013856(n) = A009964(A016885(n)). (End)

A154614 Triangle read by rows where T(m,n) = m*n + m + n - 1, 1<=n<=m.

Original entry on oeis.org

2, 4, 7, 6, 10, 14, 8, 13, 18, 23, 10, 16, 22, 28, 34, 12, 19, 26, 33, 40, 47, 14, 22, 30, 38, 46, 54, 62, 16, 25, 34, 43, 52, 61, 70, 79, 18, 28, 38, 48, 58, 68, 78, 88, 98, 20, 31, 42, 53, 64, 75, 86, 97, 108, 119, 22, 34, 46, 58, 70, 82, 94, 106, 118, 130, 142

Offset: 1

Vincenzo Librandi, Jan 16 2009

T(m,n)+2 = (n+1)*(m+1) is not prime.
T(m,m)+2 = (m+1)^2.
First column: A005843; second column: A112414; third column: 2*A020742; fourth column: A016885. - Vincenzo Librandi, Nov 17 2012

			Triangle begins:
2;
4, 7;
6, 10, 14;
8, 13, 18, 23;
10, 16, 22, 28, 34;
12, 19, 26, 33, 40, 47;
14, 22, 30, 38, 46, 54, 62;
16, 25, 34, 43, 52, 61, 70, 79;
18, 28, 38, 48, 58, 68, 78, 88, 98;
20, 31, 42, 53, 64, 75, 86, 97, 108, 119; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A040976, A005843, A112414, A020742, A016885.

[(n*k + n + k - 1): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 17 2012

t[n_,k_]:=n*k + n + k - 1; Table[t[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 17 2012 *)

A186041 Numbers of the form 3k + 2, 5k + 3, or 7*k + 4.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 62, 63, 65, 67, 68, 71, 73, 74, 77, 78, 80, 81, 83, 86, 88, 89, 92, 93, 95, 98, 101, 102, 103, 104, 107, 108, 109, 110, 113, 116, 118, 119, 122

Offset: 1

Klaus Brockhaus, Feb 11 2011, Mar 09 2011

n is in the sequence iff n is in A016789 or in A016885 or in A017029.
First differences are periodic with period length 57. Least common multiple of 3, 5, 7 is 105; number of terms <= 105 is 57.
Sequence is not essentially the same as A053726: a(n) = A053726(n-3) for 3 < n < 33, a(34)=62, A053726(34-3)=61.
Sequence is not essentially the same as A104275: a(n) = A104275(n-2) for 3 < n < 33, a(34)=62, A104275(34-3)=61.

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

Cf. A016789, A016885, A017029, A053726, A104275.

IsA186041:=func< n | exists{ k: k in [0..n div 3] | n in [3*k+2, 5*k+3, 7*k+4] } >; [ n: n in [1..200] | IsA186041(n) ];

Take[With[{no=50},Union[Join[3Range[0,no]+2,5Range[0,no]+3,7Range[0,no]+4]]],70]  (* Harvey P. Dale, Feb 16 2011 *)

a(n) = a(n-57) + 105.
a(n) = a(n-1) + a(n-57) - a(n-58).
G.f.: x*(x^57 + x^56 + x^55 + x^54 + 3*x^53 + 3*x^52 + 2*x^51 + x^50 + 3*x^49 + x^48 + 2*x^47 + 3*x^46 + 2*x^45 + x^44 + 2*x^43 + x^42 + 3*x^41 + x^40 + 2*x^39 + 3*x^38 + x^37 + 2*x^36 + 2*x^35 + x^34 + 2*x^33 + x^32 + x^31 + 2*x^30 + 3*x^29 + 3*x^28 + 2*x^27 + x^26 + x^25 + 2*x^24 + x^23 + 2*x^22 + 2*x^21 + x^20 + 3*x^19 + 2*x^18 + x^17 + 3*x^16 + x^15 + 2*x^14 + x^13 + 2*x^12 + 3*x^11 + 2*x^10 + x^9 + 3*x^8 + x^7 + 2*x^6 + 3*x^5 + 3*x^4 + x^3 + x^2 + x + 2) / ((x - 1)^2*(x^2 + x + 1)*(x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)).

A186042 Numbers of the form 2k + 1, 3k + 2, or 5*k + 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 78, 79, 80, 81, 83, 85, 86, 87, 88, 89, 91, 92, 93, 95, 97

Offset: 1

Klaus Brockhaus, Feb 11 2011, Mar 09 2011

n is in the sequence iff n is in A005408 or in A016789 or in A016885.

First differences are periodic with period length 22. Least common multiple of 2, 3, 5 is 30; number of terms <= 30 is 22.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1).

Cf. A005408, A016789, A016885.

IsA186042:=func< n | exists{ k: k in [0..n div 2] | n in [2*k+1, 3*k+2, 5*k+3] } >; [ n: n in [1..100] | IsA186042(n) ];

LinearRecurrence[{2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1},{1,2,3,5,7,8,9,11,13,14,15,17,18,19,20,21,23,25,26,27,28,29},71] (* Ray Chandler, Jul 12 2015 *)

isok(n) = (n % 2) || ((n % 3)==2) || ((n % 5)==3); \\ Michel Marcus, Jul 26 2017

a(n) = a(n-22) + 30.
a(n) = a(n-1) + a(n-22) - a(n-23).
G.f.: x*(x^21 + x^19 + x^17 + x^16 + x^15 + x^13 + x^11 + x^10 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + 1) / ((x - 1)^2*(x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)).

A279895 a(n) = n(5n + 11)/2.

Original entry on oeis.org

0, 8, 21, 39, 62, 90, 123, 161, 204, 252, 305, 363, 426, 494, 567, 645, 728, 816, 909, 1007, 1110, 1218, 1331, 1449, 1572, 1700, 1833, 1971, 2114, 2262, 2415, 2573, 2736, 2904, 3077, 3255, 3438, 3626, 3819, 4017, 4220, 4428, 4641, 4859, 5082, 5310, 5543, 5781, 6024, 6272, 6525

Offset: 0

Bruno Berselli, Dec 22 2016

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566, A008589, A017281.
Second bisection of A165720.
The first differences are in A016885.
Cf. similar sequences provided by P(s,m)+s*m, where P(s,m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number: A008585 (s=2), A055999 (s=3), A028347 (s=4), A140091 (s=5), A033537 (s=6), this sequence (s=7), A067725 (s=8).

Magma
```
[n*(5*n+11)/2: n in [0..60]];
```

Table[n (5 n + 11)/2, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{0,8,21},60] (* Harvey P. Dale, Nov 14 2022 *)

PARI
```
vector(60, n, n--; n*(5*n+11)/2)
```
Python
```
[n*(5*n+11)/2 for n in range(60)]
```
Sage
```
[n*(5*n+11)/2 for n in range(60)]
```

O.g.f.: x*(8 - 3*x)/(1 - x)^3.
E.g.f.: x*(16 + 5*x)*exp(x)/2.
a(n+h) - a(n-h) = h*A017281(n+1), with h>=0. A particular case:
a(n) - a(-n) = 11*n = A008593(n).
a(n+h) + a(n-h) = 2*a(n) + A033429(h), with h>=0. A particular case:
a(n) + a(-n) = A033429(n).
a(n) - a(n-2) = A017281(n) for n>1. Also:
40*a(n) + 121 = A017281(n+1)^2.
a(n) = A000566(n) + 7*n, also a(n) = A000566(n) + A008589(n). - Michel Marcus, Dec 22 2016

A319524 a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + mprime(n+1) and prime(n+1) + mprime(n+2), m >= 1, n >= 1.

Original entry on oeis.org

8, 33, 40, 128, 115, 302, 226, 226, 835, 401, 734, 1718, 1030, 842, 3121, 3475, 1401, 2339, 5108, 1969, 3233, 2486, 6491, 9692, 10298, 5560, 11552, 6211, 4177, 7987, 6022, 18763, 16678, 21893, 8001, 25585, 13523, 9682, 30961, 32035, 7057, 36089, 19105, 39002, 7162, 47041, 50163, 51752

Offset: 1

Alexandra Hercilia Pereira Silva, Sep 22 2018

Construct a table T in which T(m,n) = prime(n) + m*prime(n+1) as shown below. Then a(n) is defined as the smallest number appearing both in column n and column n+1, so a(1)=8, a(2)=33, a(3)=40, etc.
.
   m\n|  1     2     3     4     5     6     7     8  ...
  ----+--------------------------------------------------
    1 |  5   --8    12    18    24    30    36    42  ...
      |
    2 |  8--  13    19    29    37    47    55    65  ...
      |
    3 | 11    18    26    40    50    64    74    88  ...
      |                  /
    4 | 14    23    33  / 51    63    81    93   111  ...
      |            /   /
    5 | 17    28  / 40-   62    76    98   112   134  ...
      |          /
    6 | 20    33-   47    73    89   115   131   157  ...
      |                             /
    7 | 23    38    54    84   102 / 132   150   180  ...
      |                           /
    8 | 26    43    61    95   115   149   169   203  ...
      |
    9 | 29    48    68   106   128   166   188   226  ...
      |                       /                 /
   10 | 32    53    75   117 / 141   183   207 / 249  ...
      |                     /                 /
   11 | 35    58    82   128   154   200   226   272  ...
      |
   12 | 38    63    89   139   167   217   245   295  ...
      |
   13 | 41    68    96   150   180   234   264   318  ...
      |
   14 | 44    73   103   161   193   251   283   341  ...
      |
   15 | 47    78   110   172   206   268   302   364  ...
      |                                   /
   16 | 50    83   117   183   219   285 / 321   387  ...
      |                                 /
   17 | 53    88   124   194   232   302   340   410  ...
      |
  ... |...   ...   ...   ...   ...   ...   ...   ...  ...
Conjectures:
1. There are infinitely many pairs of consecutive equal terms. (Note that the first pair is (a(7), a(8)).)
2. There exists no N such that the sequence is monotonic for n > N.
From Amiram Eldar, Sep 22 2018: (Start)
Theorem 1: The intersection of the two mentioned arithmetic progressions is always nonempty.
Corollary: The sequence is infinite. (End)
Sequences that derive from this:
1. Positions in {s(n)} at which a(n) occurs: (2,6,5,11,8,17,19,...).
2. Positions in {s(n+1)} at which a(n) occurs: (1,4,3,9,6,15,15,...).
3. Differences between these two sequences: (1,2,2,2,2,4,...).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 600 terms from Muniru A Asiru)
Fourth International contest of logical problems, Problem 7, the Ludomind Society.
Fifth International contest of logical problems, Problem 6, the Ludomind Society, 2009.
Olivier Gérard, in reply to Zak Seidov, 11 related sequences, SeqFan list, Apr 14 2016.

Cf. A001043, A016789, A016885, A017041, A017473, A269100.

P:=Filtered([1..10000],IsPrime);;
T:=List([1..Length(P)-1],n->List([1..Length(P)-1],m->P[n]+m*P[n+1]));;
a:=List([1..50],k->Minimum(List([1..Length(T)-1],i->Intersection(T[i],T[i+1]))[k])); # Muniru A Asiru, Sep 26 2018

a[n_]:=ChineseRemainder[{Prime[n],Prime[n+1]},{Prime[n+1],Prime[n+2]} ];Array[a,44] (* Amiram Eldar, Sep 22 2018 *)

Table from Jon E. Schoenfield, Sep 23 2018

More terms from Amiram Eldar, Sep 22 2018

A354935 Row 5 of A354940: Numbers k for which A345992(k) = 5, divided by 5.

Original entry on oeis.org

3, 6, 8, 11, 13, 16, 23, 26, 31, 36, 41, 43, 46, 51, 53, 56, 61, 71, 73, 81, 83, 86, 91, 96, 101, 103, 106, 113, 116, 121, 128, 131, 141, 146, 151, 163, 166, 171, 173, 176, 181, 191, 193, 196, 206, 211, 223, 226, 233, 241, 243, 251, 256, 263, 271, 276, 281, 283, 293, 301, 311, 313, 321, 326, 331, 343, 346, 353, 356, 361

Offset: 1

Antti Karttunen, Jun 15 2022

nonn

Apparently, all terms are either of the form 5k+1 (in A016861), or of the form 5k+3 (in A016885).

Row 5 of A354940.

Cf. A016861, A016885, A344005, A345992.

q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 5*n], m++]; GCD[5*n, m] == 5]; Select[Range[360], q] (* Amiram Eldar, Jun 17 2022 *)

PARI
```
A354935(n) = A354940sq(5,n);
```

cp's OEIS Frontend

A321483 a(n) = 7*2^n + (-1)^n.

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2*n - 2*k*n + 2*n^2, with 0 <= k < n.

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A013856 a(n) = 10^(5*n + 3).

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A013896 a(n) = 20^(5*n + 3).

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A154614 Triangle read by rows where T(m,n) = m*n + m + n - 1, 1<=n<=m.

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A186041 Numbers of the form 3*k + 2, 5*k + 3, or 7*k + 4.

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A186042 Numbers of the form 2*k + 1, 3*k + 2, or 5*k + 3.

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A279895 a(n) = n*(5*n + 11)/2.

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

A186041 Numbers of the form 3k + 2, 5k + 3, or 7*k + 4.

A186042 Numbers of the form 2k + 1, 3k + 2, or 5*k + 3.

A279895 a(n) = n(5n + 11)/2.

A319524 a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + mprime(n+1) and prime(n+1) + mprime(n+2), m >= 1, n >= 1.