A126665 -id:A126665 - cp's OEIS frontend

A126719 a(n) = -n^2 + 9*n + 23.

23, 31, 37, 41, 43, 43, 41, 37, 31, 23, 13, 1, -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

Offset: 0

Michael M. Ross, Mar 13 2007

Derivation is similar to that of A126665, which see for further information.

			For n=8, -1*8^2 + 9*8 + 23 = 31.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Michael M. Ross, Natural Numbers.
Robert Sacks, Number Spiral: Method of Common Differences.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A126665.

A126719:=n->-n^2+9*n+23: seq(A126719(n), n=0..100); # Wesley Ivan Hurt, Jan 20 2017

Table[-n^2+9n+23,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{23,31,37},60] (* Harvey P. Dale, Oct 19 2011 *)

a(n)=-n^2+9*n+23 \\ Charles R Greathouse IV, Oct 07 2015

From Harvey P. Dale, Oct 19 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=23, a(1)=31, a(2)=37.
G.f.: ((38 - 13*x)*x - 23)/(x-1)^3. (End)
E.g.f.: exp(x)*(23 + 8*x - x^2). - Elmo R. Oliveira, Nov 02 2024

A186950 a(n) = n^2 - 47*n + 479.

Original entry on oeis.org

479, 433, 389, 347, 307, 269, 233, 199, 167, 137, 109, 83, 59, 37, 17, -1, -17, -31, -43, -53, -61, -67, -71, -73, -73, -71, -67, -61, -53, -43, -31, -17, -1, 17, 37, 59, 83, 109, 137, 167, 199, 233, 269, 307, 347, 389, 433, 479, 527, 577, 629, 683, 739, 797, 857

Offset: 0

Arkadiusz Wesolowski, Mar 01 2011

a(n) are distinct primes for 0 <= n <= 14. There are 22 distinct (positive and negative) values of primes between a(0) = 479 and a(48) = 527.

For n < 15 and n > 32, the prime numbers of this sequence are in A059425. - Bruno Berselli, Mar 04 2011

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..1000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 479.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A059425, A126665.

[n^2-47*n+479 : n in [0..53]]; // Arkadiusz Wesolowski, Mar 05 2011

seq(n^2-47*n+479, n=0..53); # Arkadiusz Wesolowski, Mar 08 2011

Table[n^2 - 47*n + 479, {n, 0, 53}] (* Arkadiusz Wesolowski, Mar 05 2011 *)
LinearRecurrence[{3,-3,1},{479,433,389},60] (* Harvey P. Dale, Nov 28 2018 *)

for(n=0, 53, print1(n^2-47*n+479, ", ")); \\ Arkadiusz Wesolowski, Mar 02 2011

G.f.: (479 - 1004*x + 527*x^2)/(1-x)^3. - Bruno Berselli, Mar 05 2011
a(n+19) = -A126665(n). - Arkadiusz Wesolowski, Oct 24 2013
From Elmo R. Oliveira, Nov 02 2024: (Start)
E.g.f.: (479 - 46*x + x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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.

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

cp's OEIS Frontend

A126719 a(n) = -n^2 + 9*n + 23.

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A186950 a(n) = n^2 - 47*n + 479.

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

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Magma

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PARI

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

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