A027753 -id:A027753 - cp's OEIS frontend

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, -419, -1321, -2129, -2843, -3463, -3989, -4421, -4759, -5003, -5153, -5209, -5171, -5039, -4813, -4493, -4079, -3571, -2969, -2273, -1483, -599, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387

Offset: 1

Eric W. Weisstein

Terms are listed in the order of their appearance in sequence b.

This is a transformed version of the polynomial P(x) = 47*x^2 + 9*x - 5209 whose absolute value gives 43 distinct primes for -24 <= x <= 18, found by G. W. Fung in 1988. - Hugo Pfoertner, Dec 13 2019

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004 (ISBN 0-387-20860-7); see Section A17, p. 59.
Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004. See p. 147.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles & Problems Connection.
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A002383, A005471, A005846, A007635, A022464, A027753, A027755, A027758, A048059, A050267, A050268, A116206, A117081, A267252.

lst={};Do[p=47*n^2-1701*n+10181;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Table[47n^2-1701n+10181,{n,0,50}],PrimeQ] (* Harvey P. Dale, Oct 03 2011 *)

[n | n <- apply(m->47*m^2-1701*m+10181, [0..100]), isprime(abs(n))] \\ Charles R Greathouse IV, Jun 18 2017

Edited by N. J. A. Sloane, May 10 2007
Further edited by Klaus Brockhaus, Mar 20 2010
More terms (to distinguish from quadratic) from Charles R Greathouse IV, Jun 18 2017

A027752 Numbers k such that k^2 + k + 3 is prime.

Original entry on oeis.org

0, 1, 4, 7, 10, 19, 22, 25, 34, 37, 55, 70, 79, 94, 100, 112, 130, 139, 154, 157, 160, 172, 175, 187, 190, 217, 220, 229, 232, 250, 259, 262, 274, 289, 295, 304, 307, 322, 325, 334, 337, 364, 367, 370, 382, 397, 400, 415, 439, 442, 472, 475, 484

Offset: 1

Patrick De Geest

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
P. De Geest, World!Of Numbers

Cf. A014556, A027753.

[n: n in [0..1000] |IsPrime(n^2+n+3)]; // Vincenzo Librandi, Nov 20 2010

A027714 Numbers k such that k^2+k+3 is a palindrome.

Original entry on oeis.org

0, 1, 2, 5, 19, 23, 60, 71, 175, 179, 184, 243, 753, 2431, 6154, 23111, 30947, 73188, 75146, 230663, 237721, 598350, 3093852, 5492899, 17605724, 18886025, 30909092, 62127180, 76675186, 177865385, 230098566, 309230287, 549199524, 589167859, 726714741

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..50
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027715, A027688, A027752, A027753, A027712, A027716.

npalQ[n_]:=Module[{c=n^2+n+3},c==IntegerReverse[c]]; Select[Range[ 0,31*10^5],npalQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2016 *)

More terms from Giovanni Resta, Aug 29 2018

A027715 Palindromes of form k^2 + k + 3.

Original entry on oeis.org

3, 5, 9, 33, 383, 555, 3663, 5115, 30803, 32223, 34043, 59295, 567765, 5912195, 37877873, 534141435, 957747759, 5356556535, 5646996465, 53205650235, 56511511565, 358023320853, 9571923291759, 30171944917103, 309961535169903, 356681959186653, 955371999173559

Offset: 1

Patrick De Geest

Palindromes h such that 4*h - 11 is a square. - Bruno Berselli, Aug 29 2018

Giovanni Resta, Table of n, a(n) for n = 1..50
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027714, A027688, A027752, A027753, A027713, A027717.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 3; Select[f@ Range[0, 10^5], palQ] (* Giovanni Resta, Aug 29 2018 *)

More terms from Giovanni Resta, Aug 29 2018

A302445 Triangle read by rows: row n gives primes of form k^2 + n - k for 0 < k < n.

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 17, 7, 13, 19, 37, 11, 29, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 13, 19, 43, 103, 17, 71, 197, 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 19, 31, 61, 109, 151, 229, 23, 41, 131, 293, 401, 23, 29, 43, 53, 79, 113, 179, 233, 263, 443

Offset: 2

Seiichi Manyama, Apr 08 2018

			  n\k|  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16
  ---+-----------------------------------------------------------------------
    2|  2;
    3|  3,  5;
    4|
    5|  5,  7, 11, 17;
    6|
    7|  7,   , 13, 19,   , 37;
    8|
    9|   , 11,   ,   , 29,   ,   ,   ;
   10|
   11| 11, 13, 17, 23, 31, 41, 53, 67, 83, 101;
   12|
   13| 13,   , 19,   ,   , 43,   ,   ,   , 103,    ,    ;
   14|
   15|   , 17,   ,   ,   ,   ,   , 71,   ,    ,    ,    ,    , 197;
   16|
   17| 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257;

Seiichi Manyama, Rows n = 2..421, flattened
Eric Weisstein's World of Mathematics, Lucky Number of Euler
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Wikipedia, Lucky numbers of Euler

Row n: A027753 (n=3), A027755 (n=5), A048059 (n=11), A007635 (n=17), A005846 (n=41).

Cf. A000040, A014556, A302826.

a:=Filtered(Flat(List([1..10],n->List([1..n],k->k^2+n-k))),IsPrime); # Muniru A Asiru, Apr 09 2018

Map[Union@ Select[#, PrimeQ] &, Table[k^2 + n - k, {n, 23}, {k, 0, n}]] // Flatten (* Michael De Vlieger, Apr 10 2018 *)

A128878 Primes of the form 47n^2 - 1701n + 10181.

Original entry on oeis.org

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387, 34057, 36821, 39679, 45677, 48817, 52051, 65927, 81307, 89561, 102647, 107197, 116579, 126337, 131357

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Apr 17 2007

nonn

Primes are given in the order in which they arise for increasing n.
Polynomial generates 22 primes for 0 <= n <= 42, i.e., for n = 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42.
If the definition is replaced by "Numbers n of the form 47*k^2 - 1701*k + 10181 such that either n or -n is a prime" we get (essentially) A050267.

			47k^2 - 1701k + 10181 = 21647 for k = 42.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, ISBN 0-387-20860-7, Section A17, page 59.

G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles and Problems Connection.

Cf. A050267, A002383, A027753, A027755, A005471, A027758, A048059, A007635, A005846, A116206, A050268, A022464.

Select[Table[47*n^2 - 1701*n + 10181, {n, 0, 100}], # > 0 && PrimeQ[#] &] (* T. D. Noe, Aug 02 2011 *)

Edited by Klaus Brockhaus, Apr 22 2007 and by N. J. A. Sloane, May 05 2007 and May 06 2007

A333040 Even numbers m such that sigma(m) < sigma(m-1).

Original entry on oeis.org

46, 106, 118, 166, 226, 274, 298, 316, 346, 358, 406, 466, 514, 526, 562, 586, 622, 694, 706, 766, 778, 826, 838, 862, 886, 946, 1006, 1114, 1126, 1156, 1186, 1198, 1282, 1306, 1366, 1396, 1426, 1486, 1522, 1546, 1576, 1594, 1618, 1702, 1726, 1756

Offset: 1

Bernard Schott, Mar 22 2020

nonn

The even terms of A333039 represent about only 7% of the data, so they are proposed in this sequence. Some of these integers are semiprimes with for example these two families:
1) m = 2*p with p prime of the form k^2+k+3 is in A027753. The first few terms are: 46, 118, 226, 766, ... but not all the integers of this form are terms; the first 3 counterexamples are 6, 10, 1018 (see examples).
2) m = 2*p with p prime = (r*s*t+1)/2 and 2A234103. The first few terms are: 106, 166, 274, 346, 358, ... but not all the integers of this form are terms; the first 3 counterexamples are 386, 898 and 958 (see examples).
There is also this subsequence of even m = 2^2*p where p prime, congruent to 34 mod 45, is in A142330. The first few terms are: 316, 1396, 1756, 2416, ... but not all the integers of this form are terms; the first counterexample that comes from the 8th term of A142330 is 5716.
Even (and odd) numbers such that sigma(m) = sigma(m-1) are in A231546.

			166 = 2*83 and 165 = 3*5*11, as 252 = sigma(166) < sigma(165) = 288, hence 166 is a term.
386 = 2*193 and 385 = 5*7*11, as 582 = sigma(386) > sigma(385)= 576, hence 386 is not a term.
766 = 2*383 where 383 = 19^2+19+3 and 765 = 3^2*5*13, as 1152 = sigma(766) < sigma(765) = 1404, hence 766 is a term.
1018 = 2*509 where 509 = 22^2+22+3, and 1017 = 3^2*113, as 1530 = sigma(1018) > sigma(1017) = 1482, hence 1018 is not a term.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A333039.
Subsequence of A333038.
Cf. A231546.

filter:= n -> numtheory:-sigma(n) < numtheory:-sigma(n-1):
select(filter, [seq(i,i=2..2000,2)]); # Robert Israel, Mar 29 2020

Select[2 * Range[1000], DivisorSigma[1, #] < DivisorSigma[1, #-1] &] (* Amiram Eldar, Mar 24 2020 *)

isok(m) = !(m%2) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 22 2020

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.

A309844 Primes of the form n^4 + n^2 + 3.

Original entry on oeis.org

3, 5, 23, 653, 10103, 83813, 160403, 234743, 280373, 1049603, 3420653, 6252503, 11319863, 52207853, 92246423, 146422103, 174913853, 221548343, 442071653, 479807123, 577224653, 607597853, 655385603, 937921253, 1222865933, 1249233683, 1387525253, 1506177293

Offset: 1

Christopher R. Madan, Aug 19 2019

Digital root of all values > 3 is 5, compare A017221.

Subset of A027753. Subset of A017221.

Cf. A037896, A182344, A182345, A049423.

a = [];
for n = 0:1e3
    x = n.^4+n.^2+3;
    if isprime(x); a = [a,x]; end;
end

f[n_] := n^4 + n^2 + 3; Select[f /@ Range[0, 200], PrimeQ] (* Amiram Eldar, Aug 24 2019 *)

from sympy import isprime
a = []
for n in range(0,1000):
    x = n**4+n**2+3
    if isprime(x):
        a.append(x)

cp's OEIS Frontend

A050267 Primes or negative values of primes in the sequence b(n) = 47*n^2 - 1701*n + 10181, n >= 0.

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A027752 Numbers k such that k^2 + k + 3 is prime.

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Magma

A027714 Numbers k such that k^2+k+3 is a palindrome.

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Mathematica

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A027715 Palindromes of form k^2 + k + 3.

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Mathematica

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A302445 Triangle read by rows: row n gives primes of form k^2 + n - k for 0 < k < n.

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GAP

Mathematica

A128878 Primes of the form 47*n^2 - 1701*n + 10181.

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Mathematica

Extensions

A333040 Even numbers m such that sigma(m) < sigma(m-1).

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Maple

Mathematica

PARI

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

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A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

A128878 Primes of the form 47n^2 - 1701n + 10181.

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.