A092749 -id:A092749 - cp's OEIS frontend

A014556 Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1.

Original entry on oeis.org

2, 3, 5, 11, 17, 41

Offset: 1

Eric W. Weisstein

Same as n such that 4n-1 is a Heegner number 1,2,3,7,11,19,43,67,163 (see A003173 and Conway and Guy's book).

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 225.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.
I. N. Herstein and I. Kaplansky, Matters Mathematical, Chelsea, NY, 2nd. ed., 1978, see p. 38.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983.

Aram Bingham, Ternary arithmetic, factorization, and the class number one problem, arXiv:2002.02059 [math.NT], 2020. See p. 8.
Hung Viet Chu, Steven J. Miller, and Joshua M. Siktar, Composite Numbers in an Arithmetic Progression, arXiv:2411.03330 [math.HO], 2024. See p. 7.
Brady Haran and Matt Parker, Caboose Numbers, Youtube video, June 2024.
Harold M. Stark, A complete determination of the complex quadratic fields of class-number one, The Michigan Mathematical Journal 14.1 (1967): 1-27.
Eric Weisstein's World of Mathematics, Lucky Number of Euler
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A000926, A003173, A092749, A117530, A117531.

A003173 = Union[Select[-NumberFieldDiscriminant[Sqrt[-#]] & /@ Range[200], NumberFieldClassNumber[Sqrt[-#]] == 1 &] /. {4 -> 1, 8 -> 2}]; a[n_] := (A003173[[n + 4]] + 1)/4; Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Jul 16 2012, after M. F. Hasler *)
Select[Range[50],AllTrue[Table[m^2-m+#,{m,0,#-1}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 12 2017 *)

is(n)=n>1 && qfbclassno(1-4*n)==1 \\ Charles R Greathouse IV, Jan 29 2013

is(p)=for(n=1,p-1, if(!isprime(n*(n-1)+p),return(0))); 1 \\ naive; Charles R Greathouse IV, Aug 26 2022

is(p)=for(n=1,sqrt(p/3)\/1, if(!isprime(n*(n-1)+p),return(0))); 1 \\ Charles R Greathouse IV, Aug 26 2022

a(n) = (A003173(n+3) + 1)/4. - M. F. Hasler, Nov 03 2008

A094749 Triangle read by rows in which the n-th row contains the least set of n successive primes whose successive difference forms an arithmetic progression with common difference 2, (successive even numbers).

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 17, 19, 23, 29, 347, 349, 353, 359, 367, 2903, 2909, 2917, 2927, 2939, 2953, 15373, 15377, 15383, 15391, 15401, 15413, 15427, 128981, 128983, 128987, 128993, 129001, 129011, 129023, 129037, 95285633, 95285639, 95285647, 95285657

Offset: 1

Amarnath Murthy, May 24 2004

The difference between the first two primes in each row does not have to be two; what is required is that the second differences between the primes in each row are all twos. - Harvey P. Dale, Aug 09 2020

			2
3 5
5 7 11
17 19 23 29
...

Cf. A016045, A092749.

Module[{prs=Prime[Range[551*10^4]],nn=9},Join[{2,3,5},Table[ SelectFirst[ Partition[ prs,n,1],Union[Differences[#,2]]=={2}&],{n,3,nn}]]// Flatten] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 09 2020 *)

More terms from David Wasserman, Jun 07 2007

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))

A164597 a(n) = the largest integer such that {the n-th prime} + k(k + 1) is prime for all k where 0 <= k <= a(n).

Original entry on oeis.org

0, 1, 3, 0, 9, 0, 15, 0, 0, 1, 0, 0, 39, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Leroy Quet, Aug 17 2009

nonn

a(n) <= the n-th prime -2, for all n.

A092749

More terms from R. J. Mathar, Sep 27 2009

cp's OEIS Frontend

A014556 Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1.

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A094749 Triangle read by rows in which the n-th row contains the least set of n successive primes whose successive difference forms an arithmetic progression with common difference 2, (successive even numbers).

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

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PARI

A164597 a(n) = the largest integer such that {the n-th prime} + k(k + 1) is prime for all k where 0 <= k <= a(n).

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