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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Eleonora Echeverri-Toro

Eleonora Echeverri-Toro's wiki page.

Eleonora Echeverri-Toro has authored 6 sequences.

A201279 a(n) = 6n^2 + 10n + 5.

Original entry on oeis.org

5, 21, 49, 89, 141, 205, 281, 369, 469, 581, 705, 841, 989, 1149, 1321, 1505, 1701, 1909, 2129, 2361, 2605, 2861, 3129, 3409, 3701, 4005, 4321, 4649, 4989, 5341, 5705, 6081, 6469, 6869, 7281, 7705, 8141, 8589, 9049, 9521, 10005, 10501, 11009, 11529, 12061
Offset: 0

Views

Author

Eleonora Echeverri-Toro, Nov 29 2011

Keywords

Comments

Numbers n where 6n-5 is a square of a number type 6n-1.
Also sequence found by reading the line from 5, in the direction 5, 21,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
The spiral mentioned above naturally appears on a "graphene" like lattice (planar net 6^3). The opposite diagonal is A080859. - Yuriy Sibirmovsky, Oct 04 2016
First differences of A048395. - Leo Tavares, Nov 24 2021 [Corrected by Omar E. Pol, Dec 26 2021]

Links

Crossrefs

Cf. A136392, A080859.
Cf. A003154, A022144, A016754, A046092, A048395.

Programs

  • Magma
    [6*n^2 + 10*n + 5: n in [0..60]]; // Vincenzo Librandi, Dec 01 2011
  • Mathematica
    LinearRecurrence[{3,-3,1},{5,21,49},50] (* Vincenzo Librandi, Dec 01 2011 *)
Table[6 n^2 + 10 n + 5, {n, 0, 44}] (* or *)
CoefficientList[Series[(1 + x) (5 + x)/(1 - x)^3, {x, 0, 44}], x] (* Michael De Vlieger, Oct 04 2016 *)
  • PARI
    a(n)=6*n^2+10*n+5 \\ Charles R Greathouse IV, Nov 29 2011

Formula

G.f.: (1+x)*(5+x)/(1-x)^3. - Colin Barker, Jan 09 2012
a(n) = 1 + A033579(n+1). - Omar E. Pol, Jul 18 2012
a(n) = (n+1)*A001844(n+1)-n*A001844(n). [Bruno Berselli, Jan 15 2013]
From Leo Tavares, Nov 24 2021: (Start)
a(n) = A003154(n+2) - A022144(n+1). See Diamond Frame Stars illustration.
a(n) = A016754(n) + A046092(n+1). (End)

A199860 Numbers k such that 6k-5 is a composite number of the form (6x-1) * (6y-1).

Original entry on oeis.org

5, 10, 15, 20, 21, 25, 30, 32, 35, 40, 43, 45, 49, 50, 54, 55, 60, 65, 66, 70, 75, 76, 80, 83, 85, 87, 89, 90, 95, 98, 100, 105, 109, 110, 112, 115, 117, 120, 125, 130, 131, 134, 135, 140, 141, 142, 145, 150, 151, 153, 155, 158, 160, 164, 165, 168, 170, 175
Offset: 1

Views

Author

Eleonora Echeverri-Toro, Nov 11 2011

Keywords

Comments

Numbers whose associate in A091300 has at least one factorization into two factors of A016969.

Examples

			n=5 is in the sequence because 6*5-5 = 25 = 5*5 with x = y = 1.
n=10 is in the sequence because 6*10-5 = 55 = 5*11 with x=1, y=2.

Crossrefs

Cf. A199859.

Programs

  • Maple
    isA016969 := proc(n)
    (n mod 6)=5 ;
end proc:
isA016921 := proc(n)
    (n mod 6)=1 ;
end proc:
isA091300 := proc(n)
    (not isprime(n)) and isA016921(n) ;
end proc:
isA199860 := proc(n)
    if isA091300(6*n-5) then
        for d in numtheory[divisors](6*n-5) minus {1} do
            if isA016969(d) and isA016969((6*n-5)/d) then
                return true;
            end if;
        end do:
        return false;
    else
        return false;
    end if;
end proc:
for n from 5 to 210 do
    if isA199860(n) then
        printf("%d,",n) ;
    end if ;
end do; # R. J. Mathar, Nov 27 2011

A199859 Numbers k such that 6k-5 is a composite number of the form (6x-5)*(6y-5) when x or y is not equal to 1 except for k=1.

Original entry on oeis.org

1, 9, 16, 23, 29, 30, 37, 42, 44, 51, 55, 58, 61, 65, 68, 72, 79, 80, 81, 86, 93, 94, 99, 100, 105, 107, 114, 118, 120, 121, 128, 130, 133, 135, 137, 142, 146, 149, 155, 156, 159, 161, 163, 170, 172, 175, 177, 180, 184, 185, 191, 192, 194, 198, 205, 211, 212
Offset: 0

Views

Author

Eleonora Echeverri-Toro, Nov 11 2011

Keywords

Comments

Numbers whose associate in A091300 has at least one nontrivial factorization into two factors of A016921.

Crossrefs

Cf. A091300.

Programs

  • Maple
    isA016921 := proc(n)
    (n mod 6)=1 ;
end proc:
isA091300 := proc(n)
    (not isprime(n)) and isA016921(n) ;
end proc:
isA199859 := proc(n)
    if n = 1 then
        return true;
    elif isA091300(6*n-5) then
        for d in numtheory[divisors](6*n-5) minus {1,6*n-5} do
            if isA016921(d) and isA016921((6*n-5)/d) then
                return true;
            end if;
        end do:
        return false;
    else
        return false;
    end if;
end proc:
for n from 1 to 210 do
    if isA199859(n) then
        printf("%d,",n) ;
    end if ;
end do; # R. J. Mathar, Nov 25 2011

A199718 Numbers k such that 6*k-5 is composite, but 6*k-1 is prime.

Original entry on oeis.org

5, 9, 10, 15, 23, 25, 29, 30, 32, 40, 42, 43, 44, 45, 49, 58, 60, 65, 70, 72, 75, 80, 85, 87, 93, 94, 95, 98, 99, 100, 107, 109, 110, 114, 117, 120, 133, 135, 137, 140, 155, 158, 159, 163, 164, 170, 172, 175, 177, 184, 185, 192, 194, 197, 198, 199, 204, 205
Offset: 1

Views

Author

Eleonora Echeverri-Toro, Nov 09 2011

Keywords

Links

Crossrefs

Cf. A186243.

Programs

  • Magma
    [ p div 6 +1: n in [4..204] | not IsPrime(p-4) and p mod 6 eq 5 where p is NthPrime(n) ]; // Bruno Berselli, Nov 09 2011
  • Mathematica
    Select[Range[2, 205], ! PrimeQ[6 # - 5] && PrimeQ[6 # - 1] &] (* T. N. Noe, Nov 09 2011 *)

A199717 Numbers k such that 6*k-1 is composite, but 6*k-5 is prime.

Original entry on oeis.org

6, 11, 13, 24, 26, 27, 31, 34, 36, 41, 46, 48, 56, 57, 62, 63, 69, 71, 73, 88, 91, 92, 96, 97, 101, 102, 104, 106, 111, 116, 119, 122, 123, 126, 132, 136, 139, 154, 166, 167, 171, 173, 174, 176, 178, 179, 187, 188, 189, 193, 196, 201, 206, 207, 209, 216, 221
Offset: 1

Views

Author

Eleonora Echeverri-Toro, Nov 09 2011

Keywords

Links

Crossrefs

Cf. A186243.

Programs

  • Magma
    [ p div 6 +1: p in PrimesUpTo(1326) | not IsPrime(p+4) and p mod 6 eq 1 ]; // Bruno Berselli, Nov 09 2011
  • Mathematica
    Select[Range[221], PrimeQ[6 # - 5] && ! PrimeQ[6 # - 1] &] (* T. D. Noe, Nov 09 2011 *)
  • PARI
    is(n)=isprime(6*n-5) && !isprime(6*n-1) \\ Charles R Greathouse IV, Jun 13 2017

A199716 Numbers k such that 6k-5 and 6k-1 are both composite.

Original entry on oeis.org

16, 20, 21, 35, 37, 50, 51, 54, 55, 61, 66, 68, 76, 79, 81, 83, 86, 89, 90, 105, 112, 115, 118, 121, 125, 128, 130, 131, 134, 141, 142, 145, 146, 149, 150, 151, 153, 156, 160, 161, 165, 168, 180, 181, 186, 190, 191, 195, 200, 202, 208, 211, 212, 219, 223
Offset: 1

Views

Author

Eleonora Echeverri-Toro, Nov 09 2011

Keywords

Crossrefs

Cf. A186243.

Programs

  • Magma
    [ n: n in [1..223] | not IsPrime(6*n-5) and not IsPrime(6*n-1) ]; // Bruno Berselli, Nov 09 2011
  • Mathematica
    Select[Range[223], ! PrimeQ[6#-5] && ! PrimeQ[6#-1] &] (* T. D. Noe, Nov 09 2011 *)
Select[Range[250],AllTrue[6#+{-5,-1},CompositeQ]&] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Oct 11 2018 *)
  • PARI
    for(n=1,1e3,if(!isprime(6*n-5)&&!isprime(6*n-1),print1(n", "))) \\ Charles R Greathouse IV, Nov 10 2011

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