A017041 -id:A017041 - cp's OEIS frontend

A047374 Numbers that are congruent to {4, 5} mod 7.

4, 5, 11, 12, 18, 19, 25, 26, 32, 33, 39, 40, 46, 47, 53, 54, 60, 61, 67, 68, 74, 75, 81, 82, 88, 89, 95, 96, 102, 103, 109, 110, 116, 117, 123, 124, 130, 131, 137, 138, 144, 145, 151, 152, 158, 159, 165, 166, 172

Offset: 1

N. J. A. Sloane

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

Cf. A017029, A017041.

Select[Range[200], MemberQ[{4, 5}, Mod[#, 7]] &] (* Amiram Eldar, May 07 2021 *)

a(n) = (14*n - 5*(-1)^n - 3)/4 \\ David Lovler, Sep 15 2022

G.f.: x*(4 + x + 2*x^2)/((1 + x)*(x - 1)^2). - R. J. Mathar, Dec 04 2011
a(n) = -(5/4)*(-1)^n + 7*(n-1)/2 + 11/4. - Viet Quoc Le Tran, Jun 14 2014
a(n) = (14*n - 5*(-1)^n - 3)/4. - David Lovler, Sep 15 2022
E.g.f.: 2 + ((14*x - 3)*exp(x) - 5*exp(-x))/4. - David Lovler, Sep 15 2022

A168336 a(n) = 5 + 7*floor((n-1)/2).

Original entry on oeis.org

5, 5, 12, 12, 19, 19, 26, 26, 33, 33, 40, 40, 47, 47, 54, 54, 61, 61, 68, 68, 75, 75, 82, 82, 89, 89, 96, 96, 103, 103, 110, 110, 117, 117, 124, 124, 131, 131, 138, 138, 145, 145, 152, 152, 159, 159, 166, 166, 173, 173, 180, 180, 187, 187, 194, 194, 201, 201, 208

Offset: 1

Vincenzo Librandi, Nov 23 2009

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

Cf. A017041, A168332, A168373.

[5+7*Floor((n-1)/2): n in [1..70]]; // Vincenzo Librandi, Sep 18 2013

Table[5 + 7 Floor[(n - 1)/2], {n, 60}] (* Bruno Berselli, Sep 18 2013 *)
CoefficientList[Series[(5 + 2 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 18 2013 *)

a(n) = 7*n - a(n-1) - 4, with n>1, a(1)=5.
G.f.: x*(5 + 2*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 18 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 18 2013
a(n) = (14*n - 7*(-1)^n - 1)/4. - Bruno Berselli, Sep 18 2013
a(n) = A168332(n) - 1 = A168373(n) + 1. - Bruno Berselli, Sep 18 2013
E.g.f.: (1/2)*(4 + (7*x - 4)*cosh(x) + (7*x + 3)* sinh(x)). - G. C. Greubel, Jul 18 2016

New definition by Vincenzo Librandi, Sep 18 2013

A256101 The broken eggs problem.

Original entry on oeis.org

301, 721, 1141, 1561, 1981, 2401, 2821, 3241, 3661, 4081, 4501, 4921, 5341, 5761, 6181, 6601, 7021, 7441, 7861, 8281, 8701, 9121, 9541, 9961, 10381, 10801, 11221, 11641, 12061, 12481, 12901, 13321, 13741, 14161, 14581, 15001, 15421, 15841

Offset: 1

Wolfdieter Lang, Apr 10 2015

This is a problem of byzantine mathematics appearing in the Codex Vinobonensis Phil. Gr. 65. See the Hunger-Vogel reference, p. 73, problem  86.
It also appears in the Tropfke reference on p. 640, 4.3.5.2, Die Eierfrau, as a special case of the Chinese Ta-yen rule (method of the great extension) treated on p. 636.
This is also a problem posed in the Alten et al. reference, p. 203, Aufgabe 3.1.6 (taken from Tropfke).
For the statement of the problem (in another setting) see the Cherowitzo link, where it is considered as an application of the Chinese remainder theorem.
The problem is to find all solutions of the common congruences: N congruent to 1 modulo 2, 3, 4, 5 and 6, and 0 modulo 7. For the application of the Chinese remainder theorem one first disposes of the moduli 2 and 6 (these congruences follow from the others).
The egg-selling woman had 301 eggs before they were broken according to problem 86 with this special solution in the Hunger-Vogel reference.

H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 203.
H. Hunger and K. Vogel, Ein byzantinisches Rechenbuch des 15.Jahrhunderts. 100 Aufgaben aus dem Codex Vindobonensis Phil. Gr. 65. (in Greek and German translation), Hermann Böhlaus Nachf., Wien, 1963 (Österr. Akad. d. Wiss., phil.-hist. Kl., Denkschriften, 78. Band, 2. Abhandlung), p. 73.
J. Tropfke, Geschichte der Elementarmathematik, Band 1, Arithmetik und Algebra, 4. Auflage, Walter de Gruyter, Berlin, New York , 1980, p. 640.

Adam Hugill, Table of n, a(n) for n = 1..10000
Bill Cherowitzo, Chinese remainder Theorem.
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A017041.

[420*n-119: n in [1..40]]; // Vincenzo Librandi, Apr 11 2015

A256101:=n->420*n-119: seq(A256101(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2015

CoefficientList[Series[(301 + 119 x) / (1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 11 2015 *)

terms=[]
n=50 #terms here
for i in range(1, n+1):
    ans=420*i-119
    terms.append(ans)
print(terms)
# Adam Hugill, Feb 22 2022

a(n) = 420*n-119, n >= 1, (note that 420 = 3*4*5*7, with pairwise coprime factors needed for the Chinese remainder theorem).
a(n) = 60*A017041(n-1) + 1, n >= 1.
G.f.: x*7*(43+17*x)/(1-x)^2. [Corrected by Vincenzo Librandi, Apr 11 2015]

Corrected G.f. rewritten. - Wolfdieter Lang, Apr 15 2015

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

A354937 Row 7 of A354940: Numbers k for which A345992(k) = 7, divided by 7.

Original entry on oeis.org

4, 5, 8, 11, 15, 19, 22, 25, 29, 32, 39, 43, 47, 50, 53, 57, 61, 64, 67, 71, 78, 81, 89, 92, 95, 99, 103, 106, 109, 113, 127, 131, 134, 137, 141, 151, 155, 162, 169, 173, 176, 179, 183, 190, 193, 197, 211, 218, 229, 232, 239, 243, 256, 257, 263, 267, 271, 274, 277, 281, 291, 295, 302, 309, 313, 316, 323, 337, 344

Offset: 1

Antti Karttunen, Jun 15 2022

nonn

Apparently, all terms are either of the form 7k+1 (in A016993), 7k+4 (A017029) or 7k+5 (A017041).

Row 7 of A354940.

Cf. A016993, A017029, A017041, A344005, A345992.

q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 7*n], m++]; GCD[7*n, m] == 7]; Select[Range[345], q] (* Amiram Eldar, Jun 17 2022 *)

PARI
```
A354937(n) = A354940sq(7,n);
```

A047303 Numbers that are congruent to {0, 1, 2, 3, 4, 6} mod 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76

Offset: 1

N. J. A. Sloane

Complement of A017041. - Michel Marcus, Sep 08 2015

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

Cf. A017041 (7n+5).

[n: n in [0..100] | n mod 7 in [0..4] cat [6]]; // Vincenzo Librandi, Sep 08 2015

A047303:=n->n+1+floor((n-2)/6)-ceil((n-1)/6)+floor((n-1)/6)-ceil(n/6)+floor(n/6): seq(A047303(n), n=1..100); # Wesley Ivan Hurt, Sep 07 2015

Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 4, 6}, Mod[#, 7]] &] (* Vincenzo Librandi, Sep 08 2015 *)
LinearRecurrence[{1,0,0,0,0,1,-1},{0,1,2,3,4,6,7},80] (* Harvey P. Dale, Sep 24 2016 *)

G.f.: x^2*(1+x+x^2+x^3+2*x^4+x^5) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Sep 07 2015: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
a(n) = n + 1 + floor((n-2)/6) - ceiling((n-1)/6) + floor((n-1)/6) - ceiling(n/6) + floor(n/6). (End)
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = (42*n - 51 + 3*cos(n*Pi) + 4*sqrt(3)*cos((1-4*n)*Pi/6) + 12*sin((1+2*n)*Pi/6))/36.
a(6k) = 7k-1, a(6k-1) = 7k-3, a(6k-2) = 7k-4, a(6k-3) = 7k-5, a(6k-4) = 7k-6, a(6k-5) = 7k-7. (End)

More terms from Vincenzo Librandi, Sep 08 2015

A047370 Numbers that are congruent to {2, 3, 5} mod 7.

Original entry on oeis.org

2, 3, 5, 9, 10, 12, 16, 17, 19, 23, 24, 26, 30, 31, 33, 37, 38, 40, 44, 45, 47, 51, 52, 54, 58, 59, 61, 65, 66, 68, 72, 73, 75, 79, 80, 82, 86, 87, 89, 93, 94, 96, 100, 101, 103, 107, 108, 110, 114, 115, 117, 121, 122, 124, 128, 129, 131, 135, 136, 138, 142

Offset: 1

N. J. A. Sloane, Dec 11 1999

Union of A017005, A017017 and A017041. - Michel Marcus, May 25 2014

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

Cf. A017005, A017017, A017041.

[7*Floor((n-1)/3)+2^((n-1) mod 3)+1: n in [1..50]]; // Wesley Ivan Hurt, May 25 2014

A047370:=n->7*floor((n-1)/3) + 2^((n-1) mod 3)+1; seq(A047370(n), n=1..50); # Wesley Ivan Hurt, May 25 2014

Select[Range[200], MemberQ[{2,3,5}, Mod[#,7]]&] (* or *) LinearRecurrence[ {1,0,1,-1}, {2,3,5,9}, 60] (* Harvey P. Dale, Apr 29 2013 *)
Table[7*Floor[(n - 1)/3] + 2^Mod[n - 1, 3] + 1, {n, 50}] (* Wesley Ivan Hurt, May 25 2014 *)

x='x + O('x^50); Vec(x*(2+x+2*x^2+2*x^3)/((1+x+x^2)*(x-1)^2)) \\ G. C. Greubel, Feb 21 2017

G.f.: x*(2+x+2*x^2+2*x^3)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Dec 04 2011
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4, with a(1)=2, a(2)=3, a(3)=5, a(4)=9. - Harvey P. Dale, Apr 29 2013
a(n) = 7*floor((n-1)/3)+2^((n-1) mod 3)+1. - Gary Detlefs, May 25 2014
a(n) = (1/9)*(21*n+4*sqrt(3)*sin((2*Pi*n)/3)-6*cos((2*Pi*n)/3)-12). - Alexander R. Povolotsky, May 25 2014
a(3k) = 7k-2, a(3k-1) = 7k-4, a(3k-2) = 7k-5. - Wesley Ivan Hurt, Jun 10 2016

More terms from Wesley Ivan Hurt, May 25 2014

A111367 Numbers k such that 7*k + 5 is prime.

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 18, 24, 32, 36, 38, 44, 54, 56, 62, 66, 72, 74, 84, 86, 96, 98, 102, 104, 108, 122, 126, 132, 138, 144, 152, 156, 164, 168, 174, 176, 182, 186, 188, 204, 206, 212, 218, 222, 228, 236, 242, 248, 254, 258, 266, 278, 282, 284, 294, 308, 314, 324

Offset: 1

Parthasarathy Nambi, Nov 07 2005

			k=108 is a term because 7*k + 5 = 761 is prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A017041, A045458, A111224.

[n: n in [0..100000] |IsPrime(7*n+5)] // Vincenzo Librandi, Nov 13 2010

is(n)=isprime(7*n+5) \\ Charles R Greathouse IV, Jun 13 2017

A116728 Number of permutations of length n which avoid the patterns 321, 1243, 2134.

Original entry on oeis.org

1, 2, 5, 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, 96, 103, 110, 117, 124, 131, 138, 145, 152, 159, 166, 173, 180, 187, 194, 201, 208, 215, 222, 229, 236, 243, 250, 257, 264, 271, 278, 285, 292, 299, 306, 313, 320, 327, 334, 341, 348, 355, 362, 369

Offset: 1

Lara Pudwell, Feb 26 2006

Nathaniel Johnston, Table of n, a(n) for n = 1..5000
Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017041.

t := taylor((4*x^3+2*x^2+1)*x/(x-1)^2,x,51):seq(coeff(t,x,n),n=1..50); # Nathaniel Johnston, Apr 27 2011

Vec(x*(1 + 2*x^2 + 4*x^3) / (1 - x)^2 + O(x^70)) \\ Colin Barker, Oct 24 2017

G.f.: x*(1 + 2*x^2 + 4*x^3) / (1 - x)^2.
For n >= 3, a(n) = 7*n - 16. - Franklin T. Adams-Watters, Sep 16 2006
a(n) = 2*a(n-1) - a(n-2) for n=4. - Colin Barker, Oct 24 2017
a(n) = A017041(n-3) for n > 2. - Georg Fischer, Oct 07 2018
E.g.f.: exp(x)*(7*x - 16) + 2*(x^2 + 5*x + 8). - Stefano Spezia, Oct 10 2022

A137187 Lucky numbers (A000959) which are congruent to 5 mod 7.

Original entry on oeis.org

33, 75, 159, 201, 285, 327, 495, 537, 579, 621, 831, 873, 957, 1041, 1167, 1209, 1251, 1419, 1503, 1545, 1587, 1797, 1839, 1923, 1965, 2133, 2217, 2301, 2343, 2427, 2511, 2763, 2973, 3099, 3183, 3351, 3477, 3603, 3687, 3771, 3897, 3981, 4023, 4107, 4443

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000959 and A017041.

cp's OEIS Frontend

A047374 Numbers that are congruent to {4, 5} mod 7.

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A168336 a(n) = 5 + 7*floor((n-1)/2).

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A256101 The broken eggs problem.

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

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A354937 Row 7 of A354940: Numbers k for which A345992(k) = 7, divided by 7.

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A047303 Numbers that are congruent to {0, 1, 2, 3, 4, 6} mod 7.

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A047370 Numbers that are congruent to {2, 3, 5} mod 7.

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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.