Robert Potter - cp's OEIS frontend

A259552 a(n) = (1/4)n^4 - (1/2)n^3 + (3/4)n^2 - (1/2)n + 41.

41, 43, 53, 83, 151, 281, 503, 853, 1373, 2111, 3121, 4463, 6203, 8413, 11171, 14561, 18673, 23603, 29453, 36331, 44351, 53633, 64303, 76493, 90341, 105991, 123593, 143303, 165283, 189701, 216731, 246553, 279353, 315323, 354661, 397571, 444263, 494953

Offset: 1

Robert Potter, Jun 30 2015

Empirical Observation: Reasonably productive (better than 85% in first 24 terms) prime-generating polynomial.
All integers generated by this polynomial for 0 < n <= 24 are prime with the exception of a(14) = 47*179, a(17) = 71*263, and a(20) = 47*773.
Negative and zero values of n also produce primes but they are not unique.
a(n) = (1/4)*n^4 - (1/2)*n^3 + (3/4)*n^2 - (1/2)*n + 809, for 0 < n <= 24, is also reasonably productive but produces composites at a(4), a(7), a(19) and a(20).
a(n) = (1/4)*n^4 - (1/2)*n^3 + (3/4)*n^2 - (1/2)*n + 641, for 0 < n <= 24, is also quite productive.

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A202018.

[(1/4)*n^4-(1/2)*n^3+(3/4)*n^2-(1/2)*n+41: n in [1..40]]; // Vincenzo Librandi, Jul 03 2015

A259552:=n->n^4/4-n^3/2+3*n^2/4-n/2+41: seq(A259552(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2015

f[n_] := n^4/4 - n^3/2 + 3 n^2/4 - n/2 + 41; Array[f, 38] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {41, 43, 53, 83, 151}, 38] (* Robert G. Wilson v, Jul 07 2015 *)

G.f.: x*(41 - 162*x + 248*x^2 - 162*x^3 + 41*x^4)/(1-x)^5. - Vincenzo Librandi, Jul 03 2015

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), n>5. - Wesley Ivan Hurt, Jul 09 2015

Corrected and extended by Vincenzo Librandi, Jul 03 2015

A258841 a(n) = 9n^2 - 237n + 1927.

Original entry on oeis.org

1927, 1699, 1489, 1297, 1123, 967, 829, 709, 607, 523, 457, 409, 379, 367, 373, 397, 439, 499, 577, 673, 787, 919, 1069, 1237, 1423, 1627, 1849, 2089, 2347, 2623, 2917, 3229, 3559, 3907, 4273, 4657, 5059, 5479, 5917, 6373, 6847, 7339, 7849, 8377, 8923, 9487, 10069

Offset: 0

Robert Potter, Jun 12 2015

Empirical observation. All integers generated by polynomial for 0 < n <= 37 are prime with the exception of a(26) = 43^2 and a(29) = 43*61.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A202018, A214732, A215814.

[9*n^2-237*n+1927: n in [0..50]]; // Vincenzo Librandi, Jun 22 2015

Table[9 n^2 - 237 n + 1927, {n, 0, 25}] (* Michael De Vlieger, Jun 12 2015 *)
LinearRecurrence[{3,-3,1},{1927,1699,1489},50] (* Harvey P. Dale, Oct 08 2024 *)

vector(50, n, 9*n^2 - 237*n + 1927) \\ Michel Marcus, Jun 21 2015

From Vincenzo Librandi, Jun 22 2015: (Start)
G.f.: (1927 - 4082*x + 2173*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(1927 - 228*x + 9*x^2). - Elmo R. Oliveira, Feb 09 2025

A215814 a(n) = 60516n^2 - 61008n + 2481403.

Original entry on oeis.org

2481403, 2480911, 2601451, 2843023, 3205627, 3689263, 4293931, 5019631, 5866363, 6834127, 7922923, 9132751, 10463611, 11915503, 13488427, 15182383, 16997371, 18933391, 20990443, 23168527, 25467643, 27887791, 30428971, 33091183, 35874427, 38778703, 41804011, 44950351

Offset: 0

Robert Potter, Aug 28 2012

The formula gives consecutive primes for n from 1 to 20, except n = 9.

This is the case m = 41*6 = 246 and k = 41 of the polynomial m^2*n^2 + (m^2 - 2*m)*n + (m^2*k) - (m-1).

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

Cf. A214732.

[60516*n^2-61008*n+2481403: n in [0..26]]; // Bruno Berselli, Aug 28 2012

A215814:=n->60516*n^2 - 61008*n + 2481403; seq(A215814(n), n=0..100); # Wesley Ivan Hurt, Nov 28 2013

Table[60516 n^2 - 61008 n + 2481403, {n, 0, 30}] (* Vincenzo Librandi, Aug 29 2012 *)

a(n)=60516*n^2-61008*n+2481403 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (2481403 - 4963298*x + 2602927*x^2)/(1-x)^3. - Bruno Berselli, Aug 28 2012
From Elmo R. Oliveira, Feb 09 2025: (Start)
E.g.f.: exp(x)*(2481403 - 492*x + 60516*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Offset changed from 1 to 0 and a(0) added from Vincenzo Librandi, Aug 29 2012

Gf adapted to the offset by Bruno Berselli, Aug 29 2012

A214732 a(n) = 25n^2 + 15n + 1021.

Original entry on oeis.org

1021, 1061, 1151, 1291, 1481, 1721, 2011, 2351, 2741, 3181, 3671, 4211, 4801, 5441, 6131, 6871, 7661, 8501, 9391, 10331, 11321, 12361, 13451, 14591, 15781, 17021, 18311, 19651, 21041, 22481, 23971, 25511, 27101, 28741, 30431, 32171, 33961, 35801, 37691

Offset: 0

Robert Potter, Jul 27 2012

This is the case m=5 and k=41 of the formula m^2*n^2 + (m^2 - 2*m)*n + (m^2*k) - (m-1). The most famous example is when m=1 and k=41 (Euler's generating polynomial). With k=41 the formula gives consecutive primes for m=10 and n=0..10, m=17 and n=0..10, m=86 and n=0..8. It is interesting to note that the sequences produced are all factors of the semiprimes produced by m=1, k=41. The other famous values to try for k are 5, 11 and 17 as these all produce primes up to k^2.

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

Cf. A215814.

[25*n^2+15*n+1021: n in [0..40]] // Vincenzo Librandi, Aug 29 2012

A214732:= n-> 25*n^2 +15*n +1021; seq(A214732(n), n=0..40); # G. C. Greubel, Apr 26 2021

Table[25n^2 +15n +1021, {n, 0, 40}] (* Vincenzo Librandi, Aug 29 2012 *)
LinearRecurrence[{3,-3,1},{1021,1061,1151},40] (* Harvey P. Dale, May 31 2025 *)

a(n)=25*n^2+15*n+1021 \\ Charles R Greathouse IV, Oct 25 2012

[25*n^2 +15*n +1021 for n in (0..40)] # G. C. Greubel, Apr 26 2021

G.f.: (1021-2002*x+1031*x^2)/(1-x)^3. - Bruno Berselli, Aug 28 2012

E.g.f.: (1021 + 40*x + 25*x^2)*exp(x). - G. C. Greubel, Apr 26 2021

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User: Robert Potter

A259552 a(n) = (1/4)*n^4 - (1/2)*n^3 + (3/4)*n^2 - (1/2)*n + 41.

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A258841 a(n) = 9*n^2 - 237*n + 1927.

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A215814 a(n) = 60516*n^2 - 61008*n + 2481403.

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A214732 a(n) = 25*n^2 + 15*n + 1021.

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A259552 a(n) = (1/4)n^4 - (1/2)n^3 + (3/4)n^2 - (1/2)n + 41.

A258841 a(n) = 9n^2 - 237n + 1927.

A215814 a(n) = 60516n^2 - 61008n + 2481403.

A214732 a(n) = 25n^2 + 15n + 1021.