A105772 -id:A105772 - cp's OEIS frontend

A105770 Expansion of (x^2-x+1)(4x^2+x+1) / ((1+x+x^2)*(1-x)^3).

1, 2, 7, 10, 17, 28, 37, 50, 67, 82, 101, 124, 145, 170, 199, 226, 257, 292, 325, 362, 403, 442, 485, 532, 577, 626, 679, 730, 785, 844, 901, 962, 1027, 1090, 1157, 1228, 1297, 1370, 1447, 1522, 1601, 1684, 1765, 1850, 1939, 2026, 2117, 2212, 2305, 2402, 2503

Offset: 0

Creighton Dement, Apr 18 2005

This sequence is "tesrokseq" at the link "Sequences in Context". The identity vesrok = jesrok + lesrok + tesrok holds.
Floretion Algebra Multiplication Program, FAMP Code: 4tesrokseq[ - .25'i + 1.25'j - .25'k - .25i' + 1.25j' - .25k' + 1.25'ii' + .25'jj' - .75'kk' + .75'ij' + .25'ik' + .75'ji' - .25'jk' + .25'ki' - .25'kj' + .25e] (Link to Sequences in Context contains further details on the "roktype" used).
Differs from A002522 (n^2+1) by two every third number.

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

Cf. A105771, A105772, A002522.

LinearRecurrence[{2, -1, 1, -2, 1},{1, 2, 7, 10, 17},51] (* Ray Chandler, Sep 23 2015 *)

Vec((1 - x + x^2)*(1 + x + 4*x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, May 19 2019

a(n) = n^2 + 1 + [0,0,2] (3-periodic). - Ralf Stephan, Nov 15 2010.
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4. - Colin Barker, May 19 2019
3*a(n) = 3*n^2 +5 -2*A061347(n). - R. J. Mathar, Oct 25 2022

A111250 Numbers n such that 7*n + 10 is prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 21, 27, 31, 33, 37, 39, 43, 49, 51, 57, 67, 73, 79, 81, 87, 91, 93, 109, 111, 117, 121, 133, 139, 141, 147, 157, 159, 163, 169, 177, 181, 183, 187, 193, 207, 211, 219, 223, 229, 231, 237, 241, 249, 259, 267, 271, 277, 297, 303, 319, 333, 339, 343

Offset: 1

Parthasarathy Nambi, Oct 31 2005

nonn

One less than the entry of A089033 at the same index.

			If n=117 then 7*n + 10 = 829 (prime).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A105772, A024902, A024900.

[ n: n in [0..1500] | IsPrime(7*n + 10) ] // Vincenzo Librandi, Jan 31 2011

Select[Range[400],PrimeQ[7#+10]&] (* Harvey P. Dale, Mar 25 2021 *)

for(n=1,453,if(isprime(7*n + 10),print1(n,",")))

Extended by Lambert Klasen (lambert.klasen(AT)gmx.net), Nov 02 2005

A153351 Numbers n such that 7*n+2 is not prime.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 22, 24, 25, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 49, 50, 52, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

Vincenzo Librandi, Dec 24 2008

			Distribution of the odd terms in the following triangular array:
1;
*,*;
*,*,*;
*,*,*,*;
*,*,*,*,17;
*,9,*,*,*,*;
*,*,*,19,*,*,*;
7,*,*,*,*,*,*,41;
*,*,*,*,*,35,*,*,*;
*,*,*,*,*,*,*,*,*,*;
*,*,*,*,*,*,49,*,*,*,*;
*,*,*,*,39,*,*,*,*,*,*,89;
*,19,*,*,*,*,*,*,73,*,*,*,*; etc.
where * marks the non-integer values of (4*h*k + 2*k + 2*h - 1)/7 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A105772.

[n: n in [0..150] | not IsPrime(7*n + 2)]; // Vincenzo Librandi, Jan 13 2013

Select[Range[200], !PrimeQ[7 # + 2] &] (* Vincenzo Librandi, Jan 13 2013 *)

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010

A023223 Primes p such that 7*p + 2 is also prime.

Original entry on oeis.org

3, 5, 11, 23, 47, 53, 71, 101, 107, 131, 167, 173, 197, 251, 257, 293, 311, 317, 353, 383, 431, 461, 467, 563, 587, 593, 683, 701, 773, 797, 821, 827, 863, 887, 911, 953, 977, 983, 1031, 1091, 1097, 1103, 1151, 1181, 1187, 1193, 1217, 1223, 1277, 1301, 1307, 1373

Offset: 1

David W. Wilson

nonn

Subsequence of A105772. Except for the first term all others are congruent to 5 (mod 6) because 7*(6n+1)+2 is divisible by 3. - John Cerkan, Jul 08 2016

			3 is in the sequence because 7 * 3 + 2 = 23, which is prime.
5 is in the sequence because 7 * 5 + 2 = 37, which is prime.
7 is not in the sequence because 7 * 7 + 2 = 51 = 3 * 17.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A045392, A105772.

[n: n in [0..100000] | IsPrime(n) and IsPrime(7*n+2)]; // Vincenzo Librandi, Nov 19 2010

Select[Prime[Range[250]], PrimeQ[7# + 2] &] (* Alonso del Arte, Apr 08 2015 *)

lista(nn) = forprime(p=2, nn, if(isprime(7*p+2), print1(p, ", "))); \\ Altug Alkan, Jul 08 2016

A106086 Primes p such that 7p + 2 and 2p + 7 are primes.

Original entry on oeis.org

3, 5, 11, 23, 47, 53, 71, 131, 173, 197, 251, 257, 293, 317, 383, 461, 467, 587, 593, 683, 701, 773, 797, 863, 953, 983, 1031, 1103, 1151, 1187, 1193, 1217, 1301, 1307, 1373, 1451, 1481, 1607, 1721, 1787, 2111, 2207, 2237, 2333, 2633, 2903, 3023, 3221, 3347

Offset: 1

Zak Seidov, May 07 2005

nonn

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

Cf. A105760 (2n+7 is prime), A105772 (7n+2 is prime).

Cf. A107438, A023206.

[p: p in PrimesUpTo(5000)|IsPrime(7*p+2) and IsPrime(2*p+7)] // Vincenzo Librandi, Jan 30 2011

Select[Prime[Range[220]], PrimeQ[2#+7]&&PrimeQ[7#+2]&]

More terms from Rick L. Shepherd, Jan 29 2006

A107438 Primes p such that 7p+2 or 2p+7 is prime.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 41, 47, 53, 71, 83, 101, 107, 113, 131, 137, 167, 173, 191, 197, 227, 251, 257, 281, 293, 311, 317, 347, 353, 383, 401, 431, 461, 467, 503, 521, 563, 587, 593, 641, 647, 677, 683, 701, 743, 773, 797, 821, 827, 857, 863, 887, 911, 941, 947

Offset: 1

Zak Seidov, May 26 2005

nonn

Cf. A105760 Numbers n such that (2*n + 7) is prime; A105772 Numbers n such that (7*n + 2) is prime.

Select[Prime[Range[220]], PrimeQ[2#+7]||PrimeQ[7#+2]&] (* Shepherd *)

isok(n) = isprime(n) && (isprime(7*n+2) || isprime(2*n+7)); \\ Michel Marcus, Oct 06 2013

Edited by Rick L. Shepherd, Feb 01 2006

A111249 Numbers n such that 7*n + 8 is prime.

Original entry on oeis.org

3, 5, 9, 15, 17, 27, 29, 33, 39, 47, 53, 59, 63, 65, 69, 77, 87, 89, 93, 95, 99, 105, 107, 117, 125, 129, 135, 137, 143, 149, 155, 165, 183, 185, 195, 203, 209, 213, 225, 227, 237, 243, 245, 267, 275, 285, 287, 297, 303, 305, 315, 323, 327, 329, 333, 339, 345

Offset: 1

Parthasarathy Nambi, Oct 31 2005

nonn

One less than the associated entry in A024905.

			If n=125 then 7*n + 8 = 883 (prime).

Cf. A024900, A024902, A024905, A105772.

[n: n in [0..100000] | IsPrime(7*n+8)] // Vincenzo Librandi, Nov 13 2010

A111249:=n->`if`(isprime(7*n+8), n, NULL): seq(A111249(n), n=1..10^3); # Wesley Ivan Hurt, Feb 13 2017

Select[Range[1,351,2], PrimeQ[7#+8]&] (* Harvey P. Dale, Sep 11 2016 *)

is(n)=isprime(7*n+8) \\ Charles R Greathouse IV, Feb 14 2017

A359695 Numbers k such that 29^k - 2 is prime.

Original entry on oeis.org

2, 4, 8, 14, 42, 420, 1344

Offset: 1

Arsen Vardanyan, Mar 07 2023

a(8) > 10^4, if it exists. - Amiram Eldar, Mar 10 2023
All terms in this sequence are even. - Yifan Xie, Mar 12 2023
a(8) > 5*10^4, if it exists. - Michael S. Branicky, Sep 14 2024

			4 is a term because 29^4 - 2 = 707279 is a prime number.

Cf. A087886 (29^k + 2 is prime).

Cf. A128460, A128459, A128457, A109076, A090669, A105772, A109080, (and similar others).

Select[Range[1400], PrimeQ[29^# - 2] &] (* Amiram Eldar, Mar 10 2023 *)

PARI
```
is(k) = isprime(29^k - 2);
```

cp's OEIS Frontend

A105770 Expansion of (x^2-x+1)*(4*x^2+x+1) / ((1+x+x^2)*(1-x)^3).

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A111250 Numbers n such that 7*n + 10 is prime.

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Magma

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Extensions

A153351 Numbers n such that 7*n+2 is not prime.

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Magma

Mathematica

Extensions

A023223 Primes p such that 7*p + 2 is also prime.

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Magma

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PARI

A106086 Primes p such that 7*p + 2 and 2*p + 7 are primes.

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Magma

Mathematica

Extensions

A107438 Primes p such that 7*p+2 or 2*p+7 is prime.

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

A111249 Numbers n such that 7*n + 8 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A359695 Numbers k such that 29^k - 2 is prime.

A105770 Expansion of (x^2-x+1)(4x^2+x+1) / ((1+x+x^2)*(1-x)^3).

A106086 Primes p such that 7p + 2 and 2p + 7 are primes.

A107438 Primes p such that 7p+2 or 2p+7 is prime.