A082369 -id:A082369 - cp's OEIS frontend

A132233 Primes congruent to 13 (mod 30).

13, 43, 73, 103, 163, 193, 223, 283, 313, 373, 433, 463, 523, 613, 643, 673, 733, 823, 853, 883, 1033, 1063, 1093, 1123, 1153, 1213, 1303, 1423, 1453, 1483, 1543, 1663, 1693, 1723, 1753, 1783, 1873, 1933, 1993, 2053, 2083, 2113, 2143, 2203, 2293, 2383

Offset: 1

Omar E. Pol, Aug 15 2007

Primes ending in 3 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches

Subsequence of primes of A082369. - Michel Marcus, Jan 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A039949, A082369, A132230-A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 13 ]; // Vincenzo Librandi, Aug 14 2012

select(isprime, [seq(30*i+13,i=0..1000)]); # Robert Israel, Jan 24 2016

Select[Prime[Range[1000]],MemberQ[{13},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

lista(nn) = forprime(p=2, nn, if(p % 30 == 13, print1(p, ", "))); \\ Altug Alkan, Jan 23 2016

a(n) = A158746(n)*30 + 13. - Ray Chandler, Apr 07 2009

Intersection of A030431 and A002476. - Ray Chandler, Apr 07 2009

Extended by Ray Chandler, Apr 07 2009

A267985 Numbers congruent to {7, 13} mod 30.

Original entry on oeis.org

7, 13, 37, 43, 67, 73, 97, 103, 127, 133, 157, 163, 187, 193, 217, 223, 247, 253, 277, 283, 307, 313, 337, 343, 367, 373, 397, 403, 427, 433, 457, 463, 487, 493, 517, 523, 547, 553, 577, 583, 607, 613, 637, 643, 667, 673, 697, 703, 727, 733, 757, 763

Offset: 1

Arkadiusz Wesolowski, Jan 23 2016

Union of A128471 and A082369.

For all k >= 1 the numbers 2^k - a(n) and a(n)*2^k - 1 do not form a pair of primes, where n is any positive integer.

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

Cf. A082369, A128471, A267984.

[n: n in [0..763] | n mod 30 in {7, 13}];

LinearRecurrence[{1, 1, -1}, {7, 13, 37}, 52]

Vec(x*(7 + 6*x + 17*x^2)/((1 + x)*(1 - x)^2) + O(x^53))

a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4.
G.f.: x*(7 + 6*x + 17*x^2)/((1 + x)*(1 - x)^2).
a(n) = a(n-2) + 30.
a(n) = 10*(3*n - 4) - a(n-1).
From Colin Barker, Jan 24 2016: (Start)
a(n) = (30*n-9*(-1)^n-25)/2 for n>0.
a(n) = 15*n-17 for n>0 and even.
a(n) = 15*n-8 for n odd.
(End)
E.g.f.: 17 + ((30*x - 25)*exp(x) - 9*exp(-x))/2. - David Lovler, Sep 10 2022

Comment corrected by Philippe Deléham, Nov 28 2016

A271114 Expansion of (1+x)*(2+x)/(1-x)^2.

Original entry on oeis.org

2, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325

Offset: 0

Colin Barker, Mar 31 2016

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

Cf. A016921, A270700.

Cf. A016969, A082369, A100764, A128470, A132231, A156376.

Join[{2}, LinearRecurrence[{2, -1}, {7, 13}, 100]] (* G. C. Greubel, Mar 31 2016 *)

PARI
```
Vec((1+x)*(2+x)/(1-x)^2 + O(x^70))
```

G.f.: (1+x)*(2+x)/(1-x)^2.
a(n) = A270700(n)/6.
a(n) = 6*n+1 = A016921(n) for n>0.
a(n) = 2*a(n-1)-a(n-2) for n>2.
E.g.f.: 1 + (1+6*x)*exp(x). - G. C. Greubel, Mar 31 2016
From Bruno Berselli and G. C. Greubel, Mar 31 2016: (Start)
a(5*m+1) = 30*m + 7 = A132231(m+1).
a(5*m+2) = 30*m + 13 = A082369(m+1).
a(5*m+3) = 30*m + 19 = A156376(m).
a(5*m+4) = 30*m + 25 = 5*A016969(m).
a(5*m+5) = 30*m + 31 = A128470(m+1). (End)
a(n) = A100764(n+3) for n >= 1. - Georg Fischer, Oct 30 2018

A332243 Starhex honeycomb numbers: a(n) = 13 + 60n + 60n^2.

Original entry on oeis.org

13, 133, 373, 733, 1213, 1813, 2533, 3373, 4333, 5413, 6613, 7933, 9373, 10933, 12613, 14413, 16333, 18373, 20533, 22813, 25213, 27733, 30373, 33133, 36013, 39013, 42133, 45373, 48733, 52213, 55813, 59533, 63373, 67333, 71413, 75613, 79933, 84373

Offset: 0

John Elias, Feb 07 2020

			Example: a(2) = 13 + 60*2 + 60*2^2 = 373.
Illustration of initial terms:
.                               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 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
.
.    13                         133

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.

John Elias, Illustration of Initial Terms
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003154, A003215, A062786.

Subsequence of A082369: cf. formula.

Array[12 (5 #^2 + 5 # + 1) + 1 &, 38, 0] (* Michael De Vlieger, Feb 07 2020 *)
LinearRecurrence[{3,-3,1},{13,133,373},40] (* Harvey P. Dale, Nov 18 2023 *)

A332243(n)=n*(n+1)*60+13 \\ M. F. Hasler, Jun 09 2023

a(n) = 12*(5*n*(n + 1) + 1) + 1.
From Stefano Spezia, Feb 07 2020: (Start)
O.g.f.: (13 + 94*x + 13*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(13 + 120*x + 60*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-2) for n > 2. (End)
a(n) = A082369(A001844(n)). - M. F. Hasler, Jun 09 2023

A248474 Numbers congruent to 13 or 17 mod 30.

Original entry on oeis.org

13, 17, 43, 47, 73, 77, 103, 107, 133, 137, 163, 167, 193, 197, 223, 227, 253, 257, 283, 287, 313, 317, 343, 347, 373, 377, 403, 407, 433, 437, 463, 467, 493, 497, 523, 527, 553, 557, 583, 587, 613, 617, 643, 647, 673, 677, 703, 707, 733, 737, 763, 767, 793, 797

Offset: 1

Karl V. Keller, Jr., Oct 06 2014

The combination of A082369(30*n+13) and A128468(30*n+17) is the base sequence for A140533(Primes congruent to 13 or 17 mod 30).

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A082369 (30*n+13), A128468 (30*n+17).

Cf. A039949 (Primes of the form 30n-13), A132233 (Primes congruent to 13 mod 30), A140533 (Primes congruent to 13 or 17 mod 30).

Flatten[Table[{15n - 2, 15n + 2}, {n, 1, 41, 2}]] (* Alonso del Arte, Oct 06 2014 *)

Vec(x*(13*x^2+4*x+13)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Oct 07 2014

for n in range(1,101):
  print (n*30-17),
  print (n*30-13),

From Colin Barker, Oct 07 2014: (Start)
a(n) = (-15-11*(-1)^n+30*n)/2.
a(n) = a(n-1)+a(n-2)-a(n-3).
G.f.: x*(13*x^2+4*x+13) / ((x-1)^2*(x+1)). (End)
E.g.f.: 13 + ((30*x - 15)*exp(x) - 11*exp(-x))/2. - David Lovler, Sep 10 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2*(5+sqrt(5)))+sqrt(3)-sqrt(15))*Pi / (30*(sqrt(6*(5+sqrt(5)))+sqrt(5)-1)). - Amiram Eldar, Jul 30 2024

A375644 Products of prime 7-tuples (p, p+2, p+6, p+8, p+12, p+18, p+20) where p = A022009(n).

Original entry on oeis.org

955049953, 3431222498061314276004949808559579043, 1592283266230831269579139040471681157252043, 33803301949073251156918712397722608580666560525843, 241497625199060263864928600741562805703424517481233, 742885427216897360827893410354065761987031315800313, 1822898255205545910397244861266569904193155136678473

Offset: 1

Michael De Vlieger, Aug 23 2024

nonn

Primes p in A022009 belong to 11 (mod 210), therefore a(n) is congruent to the product of residues {11, 13, 17, 19, 23, 29, 31} (mod 210), hence a(n) is congruent to 193 (mod 210).

Gaps between prime factors are {2, 4, 2, 4, 6, 2}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022009, A082369, A375645.

Map[Times @@ NextPrime[#, Range[0, 6]] &, Select[Prime@ Range[2^20], AllTrue[# + {2, 6, 8, 12, 18, 20}, PrimeQ] &]]

cp's OEIS Frontend

A132233 Primes congruent to 13 (mod 30).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A267985 Numbers congruent to {7, 13} mod 30.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A271114 Expansion of (1+x)*(2+x)/(1-x)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332243 Starhex honeycomb numbers: a(n) = 13 + 60*n + 60*n^2.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A248474 Numbers congruent to 13 or 17 mod 30.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A375644 Products of prime 7-tuples (p, p+2, p+6, p+8, p+12, p+18, p+20) where p = A022009(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A332243 Starhex honeycomb numbers: a(n) = 13 + 60n + 60n^2.