A187318 -id:A187318 - cp's OEIS frontend

A301451 Numbers congruent to {1, 7} mod 9.

1, 7, 10, 16, 19, 25, 28, 34, 37, 43, 46, 52, 55, 61, 64, 70, 73, 79, 82, 88, 91, 97, 100, 106, 109, 115, 118, 124, 127, 133, 136, 142, 145, 151, 154, 160, 163, 169, 172, 178, 181, 187, 190, 196, 199, 205, 208, 214, 217, 223, 226, 232, 235, 241, 244, 250, 253, 259, 262, 268

Offset: 1

Bruno Berselli, Mar 21 2018

First bisection of A056991, second bisection of A242660.

The squares of the terms of A174396 are the squares of this sequence.

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

Cf. A274406: numbers congruent to {0, 8} mod 9.
Cf. A193910: numbers congruent to {2, 6} mod 9.
Subsequence of A016777, A026225, A029739, A033627, A047236, A047259, A055047, A055054, A056991, A153053, A187318, A242660.

a := [1,7,10];; for n in [4..60] do a[n] := a[n-1] + a[n-2] - a[n-3]; od; a;

Magma
```
&cat [[9*n+1, 9*n+7]: n in [0..40]];
```

Table[2 (2 n - 1) + (2 n - 3 (1 - (-1)^n))/4, {n, 1, 60}]
{#+1,#+7}&/@(9*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,7,10},60] (* Harvey P. Dale, Nov 08 2020 *)

Vec(x*(1 + 6*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Mar 22 2018

[2*(2*n-1)+(2*n-3*(1-(-1)**n))/4 for n in range(1,70)]

[n for n in (1..300) if n % 9 in (1,7)]

O.g.f.: x*(1 + 6*x + 2*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (3 + 8*exp(x) - 11*exp(2*x) + 18*x*exp(2*x))*exp(-x)/4.
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 2*(2*n - 1) + (2*n - 3*(1 - (-1)^n))/4. Therefore, for n even a(n) = (9*n - 4)/2, otherwise a(n) = (9*n - 7)/2.
a(2n+1) = A017173(n). a(2n) = A017245(n-1). - R. J. Mathar, Feb 28 2019

A079131 Primes such that iterated sum-of-digits (A038194) is odd.

Original entry on oeis.org

3, 5, 7, 19, 23, 37, 41, 43, 59, 61, 73, 79, 97, 109, 113, 127, 131, 149, 151, 163, 167, 181, 199, 223, 239, 241, 257, 271, 277, 293, 307, 311, 313, 331, 347, 349, 367, 379, 383, 397, 401, 419, 421, 433, 439, 457, 487, 491, 509, 523, 541, 547, 563, 577, 599

Offset: 1

Klaus Brockhaus, Dec 28 2002

Subsequence of primes of A187318. - Michel Marcus, Jun 08 2015

Primes congruent to 1, 3, 5, 7 mod 18. - Robert Israel, Jun 08 2015

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A038194, A078403, A079130, A079132, A187318.

[p: p in PrimesUpTo(600) | p mod 18 in [1,3,5,7]]; // Vincenzo Librandi, Jun 07 2015

[a: n in [0..1000] | IsPrime(a) where a is Floor(9*n/5)]; // Vincenzo Librandi, Jun 08 2015

select(isprime, [3, seq(seq(i*18+j, j=[1,5,7]),i=0..100)]); # Robert Israel, Jun 08 2015

Select[Prime[Range[120]], OddQ[Mod[#, 9]] &] (* Bruno Berselli, Aug 31 2012 *)

forprime(p=2,600,if((p%9)%2==1,print1(p,",")))

A187349 Rank transform of the sequence floor(9n/5); complement of A187348.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20, 23, 24, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 49, 51, 54, 56, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 78, 80, 82, 85, 87, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 107, 109, 111, 113, 116, 117, 119, 121, 123, 125

Offset: 1

Clark Kimberling, Mar 08 2011

nonn

See A187224.

Cf. A187224, A187350.

seqA=Table[Floor[9n/5], {n, 1, 220}] (* A187318 *)
seqB=Table[n, {n, 1, 220}]; (* A000027 *)
jointRank[{seqA_, seqB_}]:={Flatten@Position[#1, {, 1}], Flatten@Position[#1, {, 2}]}&[Sort@Flatten[{{#1, 1}&/@seqA, {#1, 2}&/@seqB}, 1]];
limseqU=FixedPoint[jointRank[{seqA, #1[[1]]}]&, jointRank[{seqA, seqB}]][[1]] (* A187349 *)
Complement[Range[Length[seqA]], limseqU] (* A187350 *)
(* by Peter J. C. Moses, Mar 07 2011 *)

A262770 A Beatty sequence: a(n)=floor(np) where p=2cos(Pi/7)=A160389.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 19, 21, 23, 25, 27, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 46, 48, 50, 52, 54, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 109, 111, 113, 115, 117, 118, 120, 122, 124, 126, 127, 129, 131, 133, 135, 136, 138, 140, 142, 144, 145, 147, 149, 151, 153, 154, 156, 158, 160, 162, 163, 165, 167, 169, 171, 172, 174, 176, 178, 180, 181, 183, 185, 187, 189, 191

Offset: 0

Patrick D McLean, Sep 30 2015

nonn

Beatty sequence of the shorter diagonal (A160389) in a regular heptagon with sidelength 1.
Complement of Beatty sequence A262773 of the longer diagonal (A231187) in a regular heptagon with sidelength 1.
First 106 terms agree with A187318, but A187318(106)=190 while A262770(106)=191.

Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
Index entries for sequences related to Beatty sequences

Complement of A262773.

Initially agrees with A187318 (because 2*cos(Pi/7) is close to 9/5).

Table[Floor[2 n Cos[Pi/7]], {n, 0, 106}] (* Michael De Vlieger, Oct 05 2015 *)

p=roots([1,-1,-2,1])(1); a(n)=floor(p*n)

a(n) = floor(n*2*cos(Pi/7)); \\ Michel Marcus, Oct 05 2015

A272915 a(n) = n + floor(5*n/6).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 23, 25, 27, 29, 31, 33, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 66, 67, 69, 71, 73, 75, 77, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 111, 113, 115, 117, 119

Offset: 0

Bruno Berselli, Jun 15 2016

Equivalently, numbers congruent to {0, 1, 3, 5, 7, 9} mod 11.

In general, n + floor((k-1)*n/k) provides the numbers congruent to {0, 1, 3, 5, ..., 2*k-3} mod (2*k-1) for k>1.

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

Cf. similar sequences with formula n+floor((k-1)*n/k): A032766 (k=2), A047220 (k=3), A047392 (k=4), A187318 (k=5).

Magma
```
[n+Floor(5*n/6): n in [0..70]];
```
Mathematica
```
Table[n + Floor[5 n/6], {n, 0, 70}]
```
Maxima
```
makelist(n+floor(5*n/6), n, 0, 70);
```
PARI
```
vector(70, n, n--; n+floor(5*n/6))
```
Python
```
[n+int(5*n/6) for n in range(70)]
```
Sage
```
[n+floor(5*n/6) for n in range(70)];
```

G.f.: x*(1 + 2 x + 2 x^2 + 2 x^3 + 2 x^4 + 2 x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5)).
a(n) = a(n-1) + a(n-6) - a(n-7).
a(6*k + r) = 11*k + 2*r - (1 - (-1)^a(r))/2, with r = 0..5.

cp's OEIS Frontend

A301451 Numbers congruent to {1, 7} mod 9.

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A079131 Primes such that iterated sum-of-digits (A038194) is odd.

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Maple

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A187349 Rank transform of the sequence floor(9n/5); complement of A187348.

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Mathematica

A262770 A Beatty sequence: a(n)=floor(n*p) where p=2*cos(Pi/7)=A160389.

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Octave

PARI

A272915 a(n) = n + floor(5*n/6).

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Author

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Magma

Mathematica

Maxima

PARI

Python

Sage

Formula

A262770 A Beatty sequence: a(n)=floor(np) where p=2cos(Pi/7)=A160389.