A014556 -id:A014556 - cp's OEIS frontend

A027754 Numbers k such that k^2 + k + 5 is prime.

0, 1, 2, 3, 6, 7, 11, 16, 17, 18, 21, 23, 27, 31, 32, 38, 42, 48, 51, 62, 67, 72, 73, 76, 77, 83, 86, 91, 93, 97, 108, 111, 116, 121, 126, 133, 136, 137, 146, 153, 158, 163, 172, 177, 182, 188, 191, 192, 193, 202, 212, 213, 216, 223, 226, 231, 247, 248

Offset: 1

Patrick De Geest

nonn

			Since 3^2 + 3 + 5 = 17, which is prime, 3 is in the sequence.
Since 4^2 + 4 + 5 = 25 = 5^2, 4 is not in the sequence.

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

Cf. A014556, A027755.

[n: n in [0..1000] |IsPrime(n^2+n+5)] // Vincenzo Librandi, Nov 20 2010

Select[Range[0, 499], PrimeQ[#^2 + # + 5] &] (* Alonso del Arte, Nov 28 2016 *)

A028823 Numbers k such that k^2 + k + 17 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 35, 37, 38, 40, 42, 44, 45, 46, 47, 49, 53, 56, 57, 59, 60, 62, 63, 64, 70, 72, 73, 75, 76, 79, 81, 82, 86, 87, 91, 92, 95, 98, 103, 104, 108, 109, 110, 113, 114

Offset: 1

Patrick De Geest

nonn

Complement of A007636. - Michel Marcus, Jun 17 2013

			15^2 + 15 + 17 = 257, which is prime, so 15 is in the sequence.
16^2 + 16 + 17 = 289 = 17^2, so 16 is not in the sequence. Much more obviously, 17 is not in the sequence either.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A007635, A007636, A014556.

[n: n in [0..1000] |IsPrime(n^2+n+17)] // Vincenzo Librandi, Nov 19 2010

Select[Range[0, 199], PrimeQ[#^2 + # + 17] &] (* Indranil Ghosh, Mar 19 2017 *)

is(n)=isprime(n^2+n+17) \\ Charles R Greathouse IV, Feb 20 2017

from sympy import isprime
print([n for n in range(201) if isprime(n**2 + n + 17)]) # Indranil Ghosh, Mar 19 2017

A092749 a(n) is the least k such that m^2 + m + k is prime for m = 0..n.

Original entry on oeis.org

2, 3, 5, 5, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41

Offset: 0

Gabriel Cunningham (gcasey(AT)mit.edu), Apr 12 2004

From Pieter Moree (moree(AT)mpim-bonn.mpg.de), Apr 16 2004: (Start)
The numbers 2, 3, 5, 11, 17, and 41 above are the only numbers B such that m^2+m+B is prime for m=0,...,B-2; this can be proved (see Mollin's paper) and is closely related to the celebrated Rabinowitsch criterion.
Since the value of m^2+m+B is B^2 for m=B-1, one cannot possibly do better than this.
An obvious question of course is whether for given n, a(n) exists at all. This is far from obvious. Assuming the generally believed k-tuplets conjecture, the answer is yes as was shown by Andrew Granville. For a proof (which is not very difficult) see the paper by Mollin.
It is also known, due to work of Lukes, Patterson and Williams, that any further elements in the above sequence, if they exist, exceed 10^18.
(End)
George Bright conjectured that a(n) exists for every n (private communication, 1974; see Dudley). - Charles R Greathouse IV, Sep 12 2013
Least prime in a succession of primes whose difference are n consecutive even numbers. - Robert G. Wilson v, Sep 30 2013
From Altug Alkan, Oct 06 2017: (Start)
Let b_i(n) be the least k such that i*(m^2 + m) + k is prime for m = 0..n and this sequence be the b_1(n) and b_3(n) be the A256302(n). Some initial values of b_i(n) for 2 <= i <= 7 are:
b_2(n): 2, 3, 7, 7, 7, 7, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19.
b_4(n): 2, 3, 5, 5, 23, 59, 59, 59, 59, 59, 59, 59, 59, 59, 653, 653, 653, 653.
b_5(n): 2, 3, 7, 7, 7, 7, 13, 13, 13, 13, 13, 13.
b_6(n): 2, 5, 5, 7, 7, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31.
b_7(n): 2, 3, 5, 5, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17. (End)

			a(1) = 3 because 0^2 + 0 + 3 = 3 is prime and 1^2 + 1 + 3 = 5 is prime and it is the smallest number with the required properties.
a(2) = 5 because 5, 7, and 11 are primes; a(3) = 5 because 5, 7, 11, and 17 are primes; a(4) = 11 because 11, 13, 17, 23, and 31 are prime. - _Robert G. Wilson v_, Sep 30 2013

Underwood Dudley, Mathematical Cranks, MAA: Washington, DC, 1992. See pp. 62f.
R. F. Lukes, C. D. Patterson, and H. C. Williams, Numerical sieving devices: their history and some applications, Nieuw Archief Wisk. 13 (1995), pp. 113-139.

R. A. Mollin, Prime-producing quadratics, Amer. Math. Monthly 104 (1997), 529-544.

Cf. A014556.

allPrime[n_, k_] := And @@ PrimeQ[Table[m^2 + m + k, {m, 0, n}]]; Table[k = 0; While[! allPrime[n, k], k++]; k, {n, 0, 39}] (* T. D. Noe, Mar 05 2012 *)
f[n_] := Block[{p = FoldList[#1 + #2 &, 1, 2 Range@ n]}, While[ Union[ PrimeQ@ p][[1]] == False, p = p + 2]; p[[1]]]; f[0] = 2; Array[f, 40, 0] (* Robert G. Wilson v, Sep 30 2013 *)

isok(k,n) = {for (m=0, n, if (!isprime(m^2 + m + k), return(0));); return (1);}
a(n) = {my(k = 0); while(!isok(k,n), k++); k;} \\ Michel Marcus, Oct 06 2017

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

Original entry on oeis.org

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

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 *)

A253827 a(n) is the number of primes of the form x^2 + x + prime(n) for 0 <= x <=prime(n).

Original entry on oeis.org

1, 2, 4, 4, 10, 4, 16, 6, 10, 13, 14, 16, 40, 8, 26, 19, 34, 21, 36, 28, 18, 18, 34, 27, 31, 68, 16, 71, 30, 23, 37, 37, 67, 44, 54, 55, 54, 26, 65, 50, 70, 68, 79, 43, 60, 70, 52, 51, 132, 38, 60, 100, 59, 111, 114, 84, 77, 68, 78, 105, 49, 67, 124, 145, 35

Offset: 1

Michel Lagneau, Jan 16 2015

nonn

Equivalently, number of distinct primes of the form x^2 - x + prime(n) for 0 <= x <= prime(n). (The point is that x^2 + x = (x+1)^2 - (x+1), so the two forms give the same numbers. x^2 - x + prime(n) is the same for x=0 and x=1, which is why the "distinct" in the comment. - Robert Israel, Oct 09 2016)

1 <= a(n) <= prime(n)-1. a(n) = prime(n)-1 iff n is in A014556. Are there any n > 1 such that a(n) = 1? - Robert Israel, Jan 16 2015

			a(13) = 40 because prime(13) = 41 and x^2 + x + 41 generates 40 prime numbers for x = 0..41.

Michel Lagneau, Table of n, a(n) for n = 1..4000
Eric Weisstein's World of Mathematics, Lucky Number of Euler

Cf. A000040, A005846, A014556, A228123.

f:= proc(n)
local p,x;
p:= ithprime(n);
nops(select(isprime, [seq(x^2+x+p,x=0..p)]))
end proc:
seq(f(n), n=1..100); # Robert Israel, Jan 16 2015

lst={};Do[p=Prime[n];k=0;Do[If[PrimeQ[x^2+x+p],k=k+1],{x,0,p}];AppendTo[lst,k],{n,1,100}];lst
Table[With[{p=Prime[n]},Count[Table[x^2+x+p,{x,0,p}],?PrimeQ]],{n,70}] (* _Harvey P. Dale, May 27 2018 *)

a(n) = my(p=prime(n)); sum(k=0, p, isprime(subst(x^2+x+p, x, k))); \\ Michel Marcus, Jan 16 2015

A302826 a(n) is number of primes of form k^2 + n - k for 0 < k < n.

Original entry on oeis.org

1, 2, 0, 4, 0, 4, 0, 2, 0, 10, 0, 4, 0, 3, 0, 16, 0, 6, 0, 5, 0, 10, 0, 10, 0, 5, 0, 13, 0, 14, 0, 3, 0, 10, 0, 16, 0, 7, 0, 40, 0, 8, 0, 6, 0, 26, 0, 12, 0, 9, 0, 19, 0, 14, 0, 9, 0, 34, 0, 21, 0, 5, 0, 19, 0, 36, 0, 13, 0, 28, 0, 18, 0, 7, 0, 31, 0, 18, 0, 19, 0, 34, 0, 15

Offset: 2

Seiichi Manyama, Apr 14 2018

nonn

			Primes of form k^2 + n - k for 0 < k < n:
  n\k|  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16  |a(n)
  ---+------------------------------------------------------------------------+----
    2|  2;                                                                    |  1
    3|  3,  5;                                                                |  2
    4|                                                                        |  0
    5|  5,  7, 11, 17;                                                        |  4
    6|                                                                        |  0
    7|  7,   , 13, 19,   , 37;                                                |  4
    8|                                                                        |  0
    9|   , 11,   ,   , 29,   ,   ,   ;                                        |  2
   10|                                                                        |  0
   11| 11, 13, 17, 23, 31, 41, 53, 67, 83, 101;                               | 10
   12|                                                                        |  0
   13| 13,   , 19,   ,   , 43,   ,   ,   , 103,    ,    ;                     |  4
   14|                                                                        |  0
   15|   , 17,   ,   ,   ,   ,   , 71,   ,    ,    ,    ,    , 197;           |  3
   16|                                                                        |  0
   17| 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257; | 16

Seiichi Manyama, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Lucky Number of Euler
Wikipedia, Lucky numbers of Euler

Cf. A014556, A188424, A302445.

A196230 Euler primes: values of x^2 - x + k for x = 1..k-1, where k is one of Euler's "lucky" numbers 2, 3, 5, 11, 17, 41.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 97, 101, 107, 113, 127, 131, 149, 151, 173, 197, 199, 223, 227, 251, 257, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601

Offset: 1

Jonathan Sondow, Oct 29 2011

See A198245 for another sequence of "Euler primes". - N. J. A. Sloane, May 29 2022
All terms are prime numbers.
k is an Euler "lucky" number iff 4k-1 is a Heegner number 1, 2, 3, 7, 11, 19, 43, 67, 163.
See A014556 (Euler's "lucky" numbers) and A003173 (Heegner numbers) for additional references and links.

			The prime 1601 is a member because 40^2-40+41 = 1601.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 225.

Cf. A003173, A014556, A198245.

H = {2, 3, 5, 11, 17, 41}; Union[Flatten[Table[ Array[ #^2 - # + H[[k]] &, H[[k]] - 1], {k, 1, 6}]]]

A208645 Least x>0 such that x^2+x+n is not prime.

Original entry on oeis.org

2, 4, 1, 2, 1, 4, 1, 1, 1, 2, 1, 10, 1, 1, 1, 2, 1, 16, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 40, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

M. F. Hasler, Mar 03 2012

nonn

By definition, a(n)>0 for all n, and a(n)>1 if n+2 is prime.

			a(0)=2 since 1^2+1+0=2 is prime, but 2^2+2+0=6 is composite.
a(1)=4 since 1^2+1+1=2, 2^2+2+1=7 and 3^2+3+1=13 are prime, but 4^2+4+1=21 is composite.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A208936, A014556, A092749.

Cf. A117530, A117531.

lx[n_]:=Module[{x=1},While[PrimeQ[x^2+x+n],x++];x]; Array[lx, 90, 0] (* Harvey P. Dale, Aug 14 2013 *)

a(n)=for( x=1, n+3, isprime(x^2+x+n) || return(x))

A268109 Terms of A268101 without repetition.

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Jan 26 2016

Cf. A268101. Supersequence of A014556.

cp's OEIS Frontend

A027754 Numbers k such that k^2 + k + 5 is prime.

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Magma

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A028823 Numbers k such that k^2 + k + 17 is prime.

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A092749 a(n) is the least k such that m^2 + m + k is prime for m = 0..n.

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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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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

A253827 a(n) is the number of primes of the form x^2 + x + prime(n) for 0 <= x <=prime(n).

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Maple

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A302826 a(n) is number of primes of form k^2 + n - k for 0 < k < n.

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A196230 Euler primes: values of x^2 - x + k for x = 1..k-1, where k is one of Euler's "lucky" numbers 2, 3, 5, 11, 17, 41.

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