Michael M. Ross - cp's OEIS frontend

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

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

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.

A134969 List of pairs of primes that are separated by the equivalent of 2 quadratic intervals. Both primes are greater than their preceding perfect squares by the same amount, or offset. The respective perfect squares can be both odd, in which case the offset is even, or both even, in which case the offset is odd.

Original entry on oeis.org

3, 11, 5, 17, 7, 19, 13, 29, 17, 37, 23, 43, 29, 53, 43, 71, 67, 103, 71, 107, 73, 109, 97, 137

Offset: 1

Michael M. Ross, Feb 04 2008

			Prime pair 71 and 107 have an odd offset of 7 from 64 and 100:
  Interval 1: 8*8 = 64 + 7 = 71.
  Interval 2: 9*9 = 81 + 7 = 88.
  Interval 3: 10*10 = 100 + 7 = 107.
Prime pair 97 and 137 have an even offset of 16 from 81 and 121:
  Interval 1: 9*9 = 81 + 16 = 97.
  Interval 2: 10*10 = 100 + 16 = 116.
  Interval 3: 11*11 = 121 + 16 = 137.
In all cases there is a complete intermediate quadratic interval (#2).
The pair (11,83), 11-9 = 83-81 = 2, does not work because there are 6 squares in between: 16,25,36,49,64,81.

Michael M. Ross, What Are Odd Squared, or Biquadratic, Paired Primes?
Michael M. Ross, Diagram of paired primes showing how two perfect pairs partially overlap: 23 and 43 span perfect squares 25 and 36; 29 and 53 span perfect squares 36 and 49.
Michael M. Ross, VBA source code providing an Excel visualization of the odd-offset prime pairs and colocation with twin primes.

Cf. A056892.

PS1 + OfN = P1, PS3 + OfN = P2, where
  PS1 = first perfect square,
  PS3 = 3rd perfect square,
  OfN = an equal positive offset from preceding perfect square, and
  P1 and P2 = the prime pair.

A126665 a(n) = -n^2 + 9*n + 53.

Original entry on oeis.org

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, -919, -983, -1049, -1117, -1187, -1259, -1333, -1409, -1487, -1567, -1649, -1733, -1819, -1907, -1997, -2089, -2183, -2279

Offset: 0

Michael M. Ross, Mar 13 2007

Quadratic equation derived from the four primes 61, 67, 71, 73 using the method of common differences. Many of the initial terms are primes.

			For n=8, -1*8^2 + 9*8 + 53 = 61.

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. A186950.

[-n^2+9*n+53 : n in [0..46]]; // Arkadiusz Wesolowski, Oct 24 2013

Table[ - n^2 + 9*n + 53, {n, 0, 46}] (* Arkadiusz Wesolowski, Oct 24 2013 *)
LinearRecurrence[{3,-3,1},{53,61,67},60] (* Harvey P. Dale, Apr 04 2024 *)

a(n) = -n^2 + 9*n + 53 \\ Michel Marcus, Jun 30 2013

From Arkadiusz Wesolowski, Oct 24 2013: (Start)
a(n) = -A186950(n+19).
G.f.: (53 - 98*x + 43*x^2)/(1 - x)^3. (End)
From Elmo R. Oliveira, Nov 02 2024: (Start)
E.g.f.: (53 + 8*x - x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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

Original entry on oeis.org

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

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

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User: Michael M. Ross

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

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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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A126665 a(n) = -n^2 + 9*n + 53.

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

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

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

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