A126719 -id:A126719 - cp's OEIS frontend

A153422 Primes of the form n^2+15n+13.

13, 29, 47, 67, 89, 113, 139, 167, 197, 229, 263, 337, 419, 463, 509, 557, 607, 659, 769, 827, 887, 1013, 1217, 1289, 1439, 1597, 2027, 2213, 2309, 2609, 2713, 2819, 2927, 3037, 3617, 3739, 3863, 3989, 4513, 4649, 4787, 5507, 5657, 6277, 6599, 6763, 7789

Offset: 1

Vincenzo Librandi, Dec 26 2008

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

Cf. A126719.

[a: n in [0..100] | IsPrime(a) where a is n^2+15*n+13]; // Vincenzo Librandi, Jul 14 2012

Select[Table[n^2+15n+13,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Jul 14 2012 *)

2119 removed and extended by R. J. Mathar, Jan 03 2009

A127316 a(n) = 2n^2 - 4n + 73.

Original entry on oeis.org

71, 73, 79, 89, 103, 121, 143, 169, 199, 233, 271, 313, 359, 409, 463, 521, 583, 649, 719, 793, 871, 953, 1039, 1129, 1223, 1321, 1423, 1529, 1639, 1753, 1871, 1993, 2119, 2249, 2383, 2521, 2663, 2809, 2959, 3113, 3271, 3433, 3599, 3769, 3943, 4121, 4303, 4489

Offset: 1

Michael M. Ross, Mar 28 2007

Extrapolates a quadratic passing through 71, 73, and 79.

			If n=10 then 2*n^2 - 4*n + 73 = 233.

Michael M. Ross, Poly Primes: The Polynomiality of Proximate Primes.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A126665, A126719.

[2*n^2-4*n+73 : n in [1..48]]; // Arkadiusz Wesolowski, Oct 24 2013

Table[2*n^2 - 4*n + 73, {n, 48}] (* Arkadiusz Wesolowski, Oct 24 2013 *)

vector(48, n, 2*n^2-4*n+73) \\ Arkadiusz Wesolowski, Oct 24 2013

G.f.: x*(71 - 140*x + 73*x^2)/(1 - x)^3. - Arkadiusz Wesolowski, Oct 24 2013
Sum_{n>=1} 1/a(n) = 1/142 + coth(sqrt(71/2)*Pi)/(2*sqrt(142)). - Amiram Eldar, Jul 30 2024
From Elmo R. Oliveira, Nov 03 2024: (Start)
E.g.f.: exp(x)*(2*x^2 - 2*x + 73) - 73.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Extended by Charles R Greathouse IV, Jul 25 2010

A140947 Four-columned array read by rows: each row gives a series of 4 consecutive primes that share a 2nd-degree polynomial relationship and produce a positive-only integer series from the derived quadratic.

Original entry on oeis.org

17, 19, 23, 29, 41, 43, 47, 53, 79, 83, 89, 97, 227, 229, 233, 239, 347, 349, 353, 359, 349, 353, 359, 367, 379, 383, 389, 397, 439, 443, 449, 457, 569, 571, 577, 587, 641, 643, 647, 653, 673, 677, 683, 691, 677, 683, 691, 701, 1031, 1033, 1039, 1049

Offset: 1

Michael M. Ross, Jul 24 2008

These "proximate-prime polynomials" exhibit high prime densities. Of the 333 under 100000, 46 have greater than 50% prime values for the first 1000 terms. 2221 positive-only PPPs have been found under 1000000. All positive-integer PPPs have complex roots (only negative-integer PPPs, which are excluded) have real roots. The roots mostly have a real part of 1/2 or a multiple of 1/2.

			For 17, 19, 23, 29 the method of common differences produces coefficients of 1, -1 and 17 for a polynomial expression of n^2 - n + 17.

Purple Math: Finding the Next Number in a Sequence: The Method of Common Differences http://www.purplemath.com/modules/nextnumb.htm
Robert Sacks, Method of Common Differences http://www.numberspiral.com/p/common_diff.html

Michael M. Ross The High Primality of Prime-Derived Quadratic Sequences (2007)
Michael M. Ross How to Use Qtest (2007)

Cf. A126665, A126719.

Method of common differences: if (P2 - P1) - (P3 - P2) = (P3 - P2) - (P4 - P3) then polynomial is degree 2.

A144204 Array A(k,n) = (n+k-2)*(n-1) - 1 (k >= 1, n >= 1) read by antidiagonals.

Original entry on oeis.org

-1, -1, 0, -1, 1, 3, -1, 2, 5, 8, -1, 3, 7, 11, 15, -1, 4, 9, 14, 19, 24, -1, 5, 11, 17, 23, 29, 35, -1, 6, 13, 20, 27, 34, 41, 48, -1, 7, 15, 23, 31, 39, 47, 55, 63, -1, 8, 17, 26, 35, 44, 53, 62, 71, 80, -1, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, -1, 10, 21, 32, 43, 54, 65, 76, 87

Offset: 1

Jonathan Vos Post, Sep 13 2008

Arises in complete intersection threefolds,
Also can be produced as a triangle read by rows: a(n, k) = nk - (n + k). - Alonso del Arte, Jul 09 2009
Kosta: Let X be a complete intersection of two hypersurfaces F_n and F_k in the projective space P^5 of degree n and k respectively. with n=>k, such that the singularities of X are nodal and F_k is smooth. We prove that if the threefold X has at most (n+k-2)*(n-1) - 1 singular points, then it is factorial.

			From _R. J. Mathar_, Jul 10 2009: (Start)
The rows A(n,1), A(n,2), A(n,3), etc., are :
.-1...0...3...8..15..24..35..48..63..80..99.120.143.168 A067998
.-1...1...5..11..19..29..41..55..71..89.109.131.155.181 A028387
.-1...2...7..14..23..34..47..62..79..98.119.142.167.194 A008865
.-1...3...9..17..27..39..53..69..87.107.129.153.179.207 A014209
.-1...4..11..20..31..44..59..76..95.116.139.164.191.220 A028875
.-1...5..13..23..35..49..65..83.103.125.149.175.203.233 A108195
.-1...6..15..26..39..54..71..90.111.134.159.186.215.246
.-1...7..17..29..43..59..77..97.119.143.169.197.227.259
.-1...8..19..32..47..64..83.104.127.152.179.208.239.272
.-1...9..21..35..51..69..89.111.135.161.189.219.251.285
.-1..10..23..38..55..74..95.118.143.170.199.230.263.298
.-1..11..25..41..59..79.101.125.151.179.209.241.275.311
.-1..12..27..44..63..84.107.132.159.188.219.252.287.324
.-1..13..29..47..67..89.113.139.167.197.229.263.299.337 Cf. A126719.
(End)
As a triangle:
. 0
. 1, 3
. 2, 5, 8
. 3, 7, 11, 15
. 4, 9, 14, 19, 24
. 5, 11, 17, 23, 29, 35
. 6, 13, 20, 27, 34, 41, 48
. 7, 15, 23, 31, 39, 47, 55, 63
. 8, 17, 26, 35, 44, 53, 62, 71, 80

Dimitra Kosta, Factoriality of complete intersection threefolds arXiv:0808.4071 [math.AG]

Row 1 = A067998(n) for n>0. Row 2 = A028387(n) for n>0.Column 1 = -A000012(n). Column 2 = A001477. Column 3 = A005408(k). Column 4 = A016789(k+1). Column 5 = A004767(k+2). Column 6 = A016897(k+3). Column 7 = A016969(k+4). Column 8 = A017053(k+5). Column 9 = A004771(k+6). Column 10 = A017257(k+7).

Cf. A000012, A001477, A004767, A004771, A005408, A016789, A016897, A016969, A017053, A028387, A067998, A126719.

A := proc(k,n) (n+k-2)*(n-1)-1 ; end: for d from 1 to 13 do for n from 1 to d do printf("%d,",A(d-n+1,n)) ; od: od: # R. J. Mathar, Jul 10 2009

a[n_, k_] := a[n, k] = n*k - (n + k); ColumnForm[Table[a[n, k], {n, 10}, {k, n}], Center] (* Alonso del Arte, Jul 09 2009 *)

A[k,n] = (n+k-2)*(n-1) - 1.

Antidiagonal sum: Sum_{n=1..d} A(d-n+1,n) = d*(d^2-2d-1)/2 = -A110427(d). - R. J. Mathar, Jul 10 2009

Duplicate of 6th antidiagonal removed by R. J. Mathar, Jul 10 2009
Keyword:tabl added by R. J. Mathar, Jul 23 2009
Edited by N. J. A. Sloane, Sep 14 2009. There was a comment that the defining formula was wrong, but it is perfectly correct.

A155557 A proximate-prime polynomial sequence generated by 2n^2 - 2n + 53089.

Original entry on oeis.org

53089, 53093, 53101, 53113, 53129, 53149, 53173, 53201, 53233, 53269, 53309, 53353, 53401, 53453, 53509, 53569, 53633, 53701, 53773, 53849, 53929, 54013, 54101, 54193, 54289, 54389, 54493, 54601, 54713, 54829, 54949, 55073, 55201, 55333, 55469, 55609, 55753, 55901

Offset: 1

Michael M. Ross, Jan 24 2009

Sequence produces 634 primes in the first 1000 terms. (A proximate-prime polynomial is a finite polynomial equation that is derived from four successive - proximate, or neighboring - primes.)
Quadratic derived from four successive primes: 53089, 53093, 53101, 53113. Produces more primes in the first 1000 terms than any other quadratic derived from 4 successive primes under 1000000. (This includes 41, 43, 47, 53 = n^2 - n + 41, which produces 582.)
For larger ranges of n, for example n=0..10^6 or n=0..10^7, the polynomial 2*n^2 + 24*n + 144323 generates more primes than 2*n^2 - 2*n + 53089. - Mike Winkler, Oct 25 2013

			For n=14, 2*(14^2) - (2*14) + 53089 = 53453.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Natural Numbers, The High Primality of Prime-Derived Quadratic Sequences.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A126665, A126719, A127316, A140947.

[2*n^2 - 2*n + 53089: n in [1..35]]; // Vincenzo Librandi, Jul 20 2011

Table[2n^2-2n+53089,{n,30}] (* or *) LinearRecurrence[{3,-3,1},{53089,53093,53101},30] (* Harvey P. Dale, Jul 19 2011 *)

QTest: Derive, analyze and solve quadratic expressions. Generate integer sequences and determine their primality. (http://www.naturalnumbers.org/QTest-NTK.html)

a(n)=2*n^2-2*n+53089 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 - 2*n + 53089.
From Harvey P. Dale, Jul 19 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4.
G.f.: x*(53089 - 106174*x + 53089*x^2)/(1-x)^3. (End)
E.g.f.: exp(x)*(2*x^2 + 53089) - 53089. - Elmo R. Oliveira, Nov 09 2024

Edited by Charles R Greathouse IV, Jul 25 2010

A243138 a(n) = n^2 + 15*n + 13.

Original entry on oeis.org

13, 29, 47, 67, 89, 113, 139, 167, 197, 229, 263, 299, 337, 377, 419, 463, 509, 557, 607, 659, 713, 769, 827, 887, 949, 1013, 1079, 1147, 1217, 1289, 1363, 1439, 1517, 1597, 1679, 1763, 1849, 1937, 2027, 2119, 2213, 2309, 2407, 2507, 2609, 2713, 2819, 2927, 3037, 3149

Offset: 0

Vincenzo Librandi, Jun 02 2014

From Klaus Purath, Dec 13 2022: (Start)
Numbers m such that 4*m + 173 is a square.
The product of two consecutive terms belongs to the sequence, a(n)*a(n+1) = a(a(n)+n).
The prime terms in this sequence are listed in A153422. Each prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -15 (mod p). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A098847, A119412, A126719, A132759, A153422.

Magma
```
[n^2+15*n+13: n in [0..50]];
```

Table[n^2 + 15 n + 13, {n, 0, 50}] (* or *) CoefficientList[Series[(13 - 10 x - x^2)/(1 - x)^3, {x, 0, 50}], x]
LinearRecurrence[{3,-3,1},{13,29,47},50] (* Harvey P. Dale, Sep 06 2020 *)

a(n)=n^2+15*n+13 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (13 - 10*x - x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
From Klaus Purath, Dec 13 2022: (Start)
a(n) = A119412(n+2) - 13.
a(n) = A132759(n+1) - 1.
a(n) = A098847(n+1) + n. (End)
Sum_{n>=0} 1/a(n) = tan(sqrt(173)*Pi/2)*Pi/sqrt(173) + 742077303/604626139. - Amiram Eldar, Feb 14 2023
E.g.f.: (13 + 16*x + x^2)*exp(x). - Elmo R. Oliveira, Oct 18 2024

cp's OEIS Frontend

A153422 Primes of the form n^2+15n+13.

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A127316 a(n) = 2*n^2 - 4*n + 73.

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A140947 Four-columned array read by rows: each row gives a series of 4 consecutive primes that share a 2nd-degree polynomial relationship and produce a positive-only integer series from the derived quadratic.

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A144204 Array A(k,n) = (n+k-2)*(n-1) - 1 (k >= 1, n >= 1) read by antidiagonals.

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A155557 A proximate-prime polynomial sequence generated by 2*n^2 - 2*n + 53089.

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A243138 a(n) = n^2 + 15*n + 13.

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A127316 a(n) = 2n^2 - 4n + 73.

A155557 A proximate-prime polynomial sequence generated by 2n^2 - 2n + 53089.