A202018 -id:A202018 - cp's OEIS frontend

A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 73303, 73361, 73421, 73483, 3443897, 3071069, 3071137, 15949847, 76553693, 365462323, 365462399, 2204597, 9721, 1842719, 246407633, 246407719, 246407807, 246407897, 246407989

Offset: 0

M. F. Hasler, May 26 2016

nonn

All terms are prime, since this is necessary and sufficient to get a prime for x = 0.
The values given in A273770 are the number of consecutive primes obtained for x = 0, 1, 2, ....
Sequence A273595 is the subsequence of terms for which 2n+1 is prime.
For even coefficients of the linear term, the answer would always be q=2, the only choice that yields a prime for x=0 and also for x=1 if (coefficient of the linear term)+3 is prime.
The initial term a(n=0) = 41 corresponds to Euler's famous prime-generating polynomial 41+x+x^2. Some subsequent terms are equal to the primes this polynomial takes for x=1,2,3,.... This stems from the fact that adding 2 to the coefficient of the linear term is equivalent to shifting the x-variable by 1. Since here we require x >= 0, we find a reduced subset of the previous sequence of primes, missing the first one, starting with q equal to the second one. (It is known that there is no better prime-generating polynomial of this form than Euler's, see the MathWorld page and A014556. "Better" means a larger p producing p-1 primes in a row. However, the prime k-tuple conjecture suggests that there should be arbitrarily long runs of primes of this form (for much larger p), i.e., longer than 41, but certainly much less than the respective p. Therefore we speak of local maxima.)

Don Reble, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index to the OEIS, Entries related to primes produced by polynomials.

Cf. A273595, A273770.
Cf. also A002837 (n such that n^2-n+41 is prime), A007634 (n such that n^2+n+41 is composite), A005846 (primes of form n^2+n+41), A097823, A144051, A187057 ... A187060, A190800, A191456 ff.
The first line of data coincides with that of A202018, A107448, A155884 (and also A140755, A142719, except for some initial terms), which are all related.

A273756(n,p=2*n+1,L=10^(5+n\10),m=0,Q)={forprime(q=1,L, for(x=1,oo, ispseudoprime(q+p*x+x^2)&& next; x>m&& [Q=q,m=x]; break));Q}

Edited, following a remark by Don Reble, by M. F. Hasler, Jan 23 2018

a(27) corrected and more terms from Don Reble, Feb 15 2018

A297785 Decimal expansion of 4099200041/9999^3.

Original entry on oeis.org

0, 0, 4, 1, 0, 0, 4, 3, 0, 0, 4, 7, 0, 0, 5, 3, 0, 0, 6, 1, 0, 0, 7, 1, 0, 0, 8, 3, 0, 0, 9, 7, 0, 1, 1, 3, 0, 1, 3, 1, 0, 1, 5, 1, 0, 1, 7, 3, 0, 1, 9, 7, 0, 2, 2, 3, 0, 2, 5, 1, 0, 2, 8, 1, 0, 3, 1, 3, 0, 3, 4, 7, 0, 3, 8, 3, 0, 4, 2, 1, 0, 4, 6, 1, 0, 5

Offset: 0

Seiichi Manyama, Jan 05 2018

			0.0041004300470053006100710083009701130131015101730197...

Paolo Xausa, Table of n, a(n) for n = 0..10000
John D. Cook, Prime generating fractions
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A005846, A202018, A272066, A297786, A297787.

First[RealDigits[4099200041/9999^3, 10, 100, -1]] (* Paolo Xausa, Jun 16 2024 *)

Sum_{k>=0} 10^(-4*k-4)*A202018(k) = 4099200041/9999^3.

A319906 Number of prime numbers of the form k^2 + k + 41 below 10^n.

Original entry on oeis.org

0, 8, 31, 86, 221, 581, 1503, 4149, 11355, 31985, 90940, 261081, 756081, 2208197, 6483148, 19132652, 56714624, 168806741, 504209234

Offset: 1

Amiram Eldar, Oct 01 2018

			The first 8 values of k^2 + k + 41 for k = 0 to 7 are above 10 and below 100: 41, 43, 47, 53, 61, 71, 83, 97, thus a(1) = 0 and a(2) = 8.

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Justin DeBenedetto and Jeremy Rouse, A 60,000 digit prime number of the form x^2 + x + 41, arXiv preprint arXiv:1207.7291 [math.NT] (2012).
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision calculation of Hardy-Littlewood constants. [local copy with permission]
Gilbert W. Fung and Hugh C. Williams, Quadratic polynomials which have a high density of prime values, Mathematics of Computation, Vol. 55, No. 191 (1990), pp. 345-353.
G. H. Hardy and J. E. Littlewood, Some problems in "Partitio numerorum", III: On the expression of a number as a sum of primes, Acta Mathematica, Vol. 44 (1923), pp. 1-70.
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (1997), pp. 529-544.
Daniel Shanks, Calculation and applications of Epstein zeta functions, Mathematics of Computation, Vol. 29, No. 129 (1975), pp. 271—287.
M. L. Stein, S. M. Ulam and M. B. Wells, A Visual Display of Some Properties of the Distribution of Primes, The American Mathematical Monthly, Vol. 71, No. 5 (1964), pp. 516-520.
Wikipedia, Ulam spiral.

Cf. A002837, A005846, A056561, A202018, A221712, A331876.

f[n_] := n^2 + n + 41; c = 0; k = 0; a={}; Do[f1 = f[k]; While[f1 < 10^n, If[PrimeQ[f1], c++]; k++; f1 = f[k]];  AppendTo[a, c], {n, 1, 10}]; a

According to Hardy and Littlewood's Conjecture F: a(n) ~ 2 * C * 10^(n/2)/(n*log(10)), where C = 3.319773... (Hardy-Littlewood constant for x^2+x+41, A221712).

A212325 Prime-generating polynomial: a(n) = n^2 + 3*n - 167.

Original entry on oeis.org

-167, -163, -157, -149, -139, -127, -113, -97, -79, -59, -37, -13, 13, 41, 71, 103, 137, 173, 211, 251, 293, 337, 383, 431, 481, 533, 587, 643, 701, 761, 823, 887, 953, 1021, 1091, 1163, 1237, 1313, 1391, 1471, 1553, 1637, 1723, 1811, 1901, 1993, 2087, 2183, 2281, 2381

Offset: 0

Marius Coman, May 14 2012

The polynomial generates 24 primes in absolute value (23 distinct ones) in row starting from n=0 (and 42 primes in absolute value for n from 0 to 46).
The polynomial n^2 - 49*n + 431 generates the same primes in reverse order.
Note: we found in the same family of prime-generating polynomials (with the discriminant equal to 677) the polynomial 13*n^2 - 311*n + 1847 (13*n^2 - 469*n + 4217) generating 23 primes and two noncomposite numbers (in absolute value) in row starting from n=0 (1847, 1549, 1277, 1031, 811, 617, 449, 307, 191, 101, 37, -1, -13, 1, 41, 107, 199, 317, 461, 631, 827, 1049, 1297, 1571, 1871).
Note: another interesting algorithm to produce prime-generating polynomials could be N = m*n^2 + (6*m+1)*n + 8*m + 3, where m, 6*m+1 and 8*m+3 are primes. For m=7 then n=t-20 we get N = 7*t^2 - 237*t + 1999, which generates the following primes: 239, 163, 101, 53, 19, -1, -7, 1, 23, 59, 109, 173, 251 (we can see the same pattern: …, -1, -m, 1, …).

Bruno Berselli, 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. A060566 (an 80 primes generating pol.), A202018 (Euler's p.g.p.), A050268, A181963, A181973, A182409, A211773, A318791, A320772, A330363 (other p.g.p.).

[n^2+3*n-167: n in [0..47]]; // Bruno Berselli, May 18 2012

Table[n^2+3n-167,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{-167,-163,-157},50] (* Harvey P. Dale, Feb 08 2020 *)

Vec((-167+338*x-169*x^2)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

apply( {A212325(n)=(n+3)*n-167}, [0..55]) \\ M. F. Hasler, Feb 11 2025

def A212325(n=None, upto=None): return(A212325(i)for i in range(n or 0, upto or 2**63)) if upto or n is None else(n+3)*n-167 # M. F. Hasler, Feb 11 2025

G.f.: (-167 + 338*x - 169x^2)/(1-x)^3. - Bruno Berselli, May 18 2012
From Elmo R. Oliveira, Feb 10 2025: (Start)
E.g.f.: exp(x)*(-167 + 4*x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Edited by Bruno Berselli, May 18 2012

A226097 a(n) = ((-1)^n + 2n - 38)(2*n - 38) + 41.

Original entry on oeis.org

1447, 1373, 1163, 1097, 911, 853, 691, 641, 503, 461, 347, 313, 223, 197, 131, 113, 71, 61, 43, 41, 47, 53, 83, 97, 151, 173, 251, 281, 383, 421, 547, 593, 743, 797, 971, 1033, 1231, 1301, 1523, 1601, 1847, 1933, 2203, 2297, 2591, 2693, 3011, 3121, 3463, 3581, 3947

Offset: 0

Arkadiusz Wesolowski, May 26 2013

a(n) are distinct primes for n = 0 to 59.

All terms are in A202018.

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..1000
Carlos Rivera, Puzzle 782
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000040, A005846, A060566, A202018.

[((-1)^n+a)*a+41 where a is 2*n-38 : n in [0..50]];

g[n_] := 2*n - 38; f[n_] := ((-1)^n + g[n])*g[n] + 41; Table[f[n], {n, 0, 50}]
EulerP[n_] := n^2 - n + 41; f[n_] := 2*n - (3 + (-1)^n)/2; LinearRecurrence[{1, 2, -2, -1, 1}, Table[EulerP@f[n], {n, 19, 15, -1}], {0, 50}]

Vec((1447 - 74*x - 3104*x^2 + 82*x^3 + 1681*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^100)) \\ Colin Barker, Aug 14 2017

G.f.: (1447-2*x*(37+1552*x-41*x^2)+(41*x^2)^2)/((1+x)^2*(1-x)^3).
From Colin Barker, Aug 14 2017: (Start)
G.f.: (1447 - 74*x - 3104*x^2 + 82*x^3 + 1681*x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = 4*n^2 - 150*n + 1447 for n even.
a(n) = 4*n^2 - 154*n + 1523 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)

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

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

Original entry on oeis.org

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

A268101 Smallest prime p such that some polynomial of the form ax^2 - bx + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 647, 1277, 1979, 2753

Offset: 1

Arkadiusz Wesolowski, Jan 26 2016

			a(1) = 2 (a prime), x^2 + 2 gives a prime for x = 1.
a(2) = 3 (a prime), 2*x^2 + 3 gives distinct primes for x = 1 to 2.
a(4) = 5 (a prime), 2*x^2 + 5 gives distinct primes for x = 1 to 4.
a(6) = 7 (a prime), 4*x^2 + 7 gives distinct primes for x = 1 to 6.
a(10) = 11 (a prime), 2*x^2 + 11 gives distinct primes for x = 1 to 10.
a(12) = 13 (a prime), 6*x^2 + 13 gives distinct primes for x = 1 to 12.
a(16) = 17 (a prime), 6*x^2 + 17 gives distinct primes for x = 1 to 16.
a(18) = 19 (a prime), 10*x^2 + 19 gives distinct primes for x = 1 to 18.
a(22) = 23 (a prime), 3*x^2 - 3*x + 23 gives distinct primes for x = 1 to 22.
a(28) = 29 (a prime), 2*x^2 + 29 gives distinct primes for x = 1 to 28.
a(29) = 31 (a prime), 2*x^2 - 4*x + 31 gives distinct primes for x = 1 to 29.
a(40) = 41 (a prime), x^2 - x + 41 gives distinct primes for x = 1 to 40.
a(41) = 647 (a prime), abs(36*x^2 - 594*x + 647) gives distinct primes for x = 1 to 41.
a(42) = 1277 (a prime), abs(36*x^2 - 666*x + 1277) gives distinct primes for x = 1 to 42.
a(43) = 1979 (a prime), abs(36*x^2 - 738*x + 1979) gives distinct primes for x = 1 to 43.
a(44) = 2753 (a prime), abs(36*x^2 - 810*x + 2753) gives distinct primes for x = 1 to 44.

Carlos Rivera, Problem 12
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A027688, A027753, A027690, A027755, A048058, A048059, A007635, A007639, A007637, A007641, A202018, A005846, A117081, A050268, A268109.

A268513 Numbers n such that bigomega(n) = bigomega(n*(n+1)+41).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 49, 53, 59, 61, 65, 67, 71, 73, 79, 82, 83, 87, 91, 97, 101, 103, 107, 113, 121, 122, 123, 131, 137, 139, 143, 149, 151, 155, 157, 159, 161, 167, 178, 179, 181, 185, 187, 191, 193, 197, 199

Offset: 1

Zak Seidov, Feb 06 2016

nonn

			Let eu(x) = x*(x + 1) + 41 and n-AP= n-almost prime, then:
both 2 and eu(2)=47 are primes,
both 49=7*7 and eu(49)=47*53 are semiprimes,
both 574=2*7*41 and eu(574)=41*83*97 are 3-AP,
both 3484=2^2*13*67 and eu(3484)=12141781=41*43*71*97 are 4-AP,
both 54224=2^4*3389 and eu(2940296441)=43^2*61*131*199 are 5-AP,
both 506022=2*3*11^2*17*41 and eu(506022)=41*43^2*71*113*421 are 6-AP,
both 7375900=2^2*5^2*7*41*257 and eu(7375900)=41*47*53*71^2*251*421 are 7-AP,
both 151072290=2*3^4*5*41*4549 and eu(151072290)=41*47*61*83*113^2*167*1097 are 8-AP.

Zak Seidov, Table of n, a(n) for n = 1..20000

Cf. A001222, A202018.

[n: n in [2..200] | &+[d[2]: d in Factorization(n)] eq &+[d[2]: d in Factorization(n^2+n+41)] ]; // Vincenzo Librandi, Feb 08 2016

Select[Range[100], PrimeOmega[#] == PrimeOmega[# (# + 1) + 41] &]

isok(n) = bigomega(n) == bigomega(n^2+n+41); \\ Michel Marcus, Feb 07 2016

A282319 a(n) = (2097203 mod n)^2 + (2097203 mod n) + 41.

Original entry on oeis.org

41, 43, 47, 53, 53, 71, 53, 53, 71, 53, 131, 173, 61, 53, 113, 53, 281, 71, 47, 53, 347, 131, 347, 173, 53, 347, 71, 53, 151, 593, 547, 421, 461, 281, 53, 593, 83, 503, 347, 53, 197, 347, 97, 1033, 593, 347, 313, 1301, 53, 53, 1097, 1933, 2203, 71

Offset: 1

Frederic Isenmann, Feb 11 2017

This sequence gives 168 prime numbers for n=1 to 168 with 63 different primes. This formula is based on the lucky numbers of Euler.

			For n = 23, a(23) = 17^2+17+41 = 347, and 347 is prime.

Frederic Isenmann, Table of n, a(n) for n=1..168.

Cf. A005846, A202018.

Table[#^2 + # + 41 &@ Mod[2097203 , n], {n, 54}] (* Michael De Vlieger, Feb 12 2017 *)
f[n_]:=Module[{x=Mod[2097203,n]},x^2+x+41]; Array[f,60] (* Harvey P. Dale, Jul 28 2017 *)

a(n)=subst(x^2+x+41,x,2097203%n) \\ Charles R Greathouse IV, Feb 14 2017

def formul(i):
    return ((i*i+2097203)%i)*((i*i+2097203)%i)+((i*i+2097203)%i)+41
for i in range(1, 169):
    n=formul(i)
    print(n, end=", ")

cp's OEIS Frontend

A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

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A297785 Decimal expansion of 4099200041/9999^3.

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A319906 Number of prime numbers of the form k^2 + k + 41 below 10^n.

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A212325 Prime-generating polynomial: a(n) = n^2 + 3*n - 167.

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A226097 a(n) = ((-1)^n + 2*n - 38)*(2*n - 38) + 41.

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

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

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A226097 a(n) = ((-1)^n + 2n - 38)(2*n - 38) + 41.

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

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

A268101 Smallest prime p such that some polynomial of the form ax^2 - bx + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.