A003173 -id:A003173 - cp's OEIS frontend

A199743 Rounded near-integers (exp(Pi*sqrt(h)) - 744)^(1/3) where h is A003173(n+3) (Heegner numbers of the form 4p-1 where p is prime).

15, 32, 96, 960, 5280, 640320

Offset: 1

Artur Jasinski, Nov 09 2011

			a(1) =     15 because     15^3 + 744 ~ exp(Pi*sqrt(7)).
a(2) =     32 because     32^3 + 744 ~ exp(Pi*sqrt(11)).
a(3) =     96 because     96^3 + 744 ~ exp(Pi*sqrt(19)).
a(4) =    960 because    960^3 + 744 ~ exp(Pi*sqrt(43)).
a(5) =   5280 because   5280^3 + 744 ~ exp(Pi*sqrt(67)).
a(6) = 640320 because 640320^3 + 744 ~ exp(Pi*sqrt(163)).

Cf. A003173, A305500.

A267195 is a supersequence (negated).

a(n) = (-j((1 + i*sqrt(h(n))) / 2))^(1/3) where h(n) = A003173(n+3) and j(x) is the j-invariant. - Andrey Zabolotskiy, Sep 30 2021

A242216 Number of partitions of n into Heegner numbers, cf. A003173.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 9, 11, 14, 17, 21, 25, 30, 36, 42, 49, 57, 66, 76, 87, 100, 114, 129, 146, 165, 185, 207, 232, 258, 287, 318, 352, 389, 428, 471, 517, 566, 619, 676, 737, 802, 872, 947, 1027, 1112, 1203, 1300, 1402, 1512, 1628, 1751, 1882, 2020, 2167, 2322

Offset: 0

Reinhard Zumkeller, May 07 2014

nonn

Heegner numbers = A003173(1..9) = {1,2,3,7,11,19,43,67,163}.

			a(10) = #{7+3, 7+2+1, 7+1+1+1, 3+3+3+1, 3+3+2+2, 3+3+2+1+1, 3+3+4x1, 3+2+2+2+1, 3+2+2+1+1+1, 3+2+5x1, 3+7x1, 5x2, 4x2+1+1, 2+2+2+4x1, 2+2+6x1, 2+8x1, 10x1} = 17;
a(11) = #{11, 7+3+1, 7+2+2, 7+2+1+1, 7+4x1, 3+3+3+2, 3+3+3+1+1, 3+3+2+2+1, 3+3+2+1+1+1, 3+3+5x1, 3+4x2, 3+2+2+2+1+1, 3+2+2+4x1, 3+2+6x1, 3+8x1, 5x2+1, 4x2+1+1+1,2+2+2+5x1, 2+2+7x1, 2+9x1, 11x1} = 21;
a(12) = #{11+1, 7+3+2, 7+3+1+1, 7+2+2+1, 7+2+1+1+1, 7+5*1, 3+3+3+3, 3+3+3+2+1, 3+3+3+1+1+1, 3+3+2+2+2, 3+3+2+2+1+1, 3+3+2+4x1, 3+3+6x1, 3+4x2+1, 3+2+2+2+1+1+1, 3+2+2+5x1, 3+2+7x1, 3+9x1, 6x2, 5x2+1+1, 4x2+4x1, 2+2+2+6x1, 2+2+8x1, 2+10x1, 12x1} = 25.

Eric Weisstein's World of Mathematics, Heegner Number
Wikipedia, Heegner number

Cf. A242217.

a242216 = p [1,2,3,7,11,19,43,67,163] where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

[#RestrictedPartitions(n,{1, 2, 3, 7, 11, 19, 43, 67, 163}):n in [1..60]]; // Marius A. Burtea, Jun 10 2019

heegnerNums = {1, 2, 3, 7, 11, 19, 43, 67, 163};
a[n_] := Length @ IntegerPartitions[n, All, heegnerNums];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jun 10 2019 *)

A242217 Number of partitions of n into distinct Heegner numbers, cf. A003173.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3

Offset: 0

Reinhard Zumkeller, May 07 2014

nonn

Heegner numbers = A003173(1..9) = {1,2,3,7,11,19,43,67,163};
0 <= a(n) <= 3;
for n > 316: a(n) = 0; 154 = smallest number m such that a(m) = 0;
number of terms greater than 0 = 303;
sum of all terms = 512.

			a(10) = #{7+3, 7+2+1} = 2;
a(11) = #{11, 7+3+1} = 2;
a(12) = #{11+1, 7+3+2} = 2;
a(13) = #{11+2, 7+3+2+1} = 2;
a(14) = #{11+3, 11+2+1} = 2;
a(15) = #{11+3+1} = 1;
a(16) = #{11+3+2} = 1;
a(17) = #{11+3+2+1} = 1;
a(18) = #{11+7} = 1;
a(19) = #{19, 11+7+1} = 2;
a(20) = #{19+1, 11+7+2} = 2;
a(316) = #{163+67+43+19+11+7+3+2+1} = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Heegner Number
Wikipedia, Heegner number

Cf. A242216.

a242217 = p [1,2,3,7,11,19,43,67,163] where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

heegnerNums = {1,2,3,7,11,19,43,67,163};
a[n_] := a[n] = Count[IntegerPartitions[n, All, heegnerNums], P_List /; Sort[P] == Union[P]];
Table[Print[n," ", a[n]]; a[n], {n,0,316}] (* Jean-François Alcover, Jun 10 2019 *)

A305500 a(n) = (-1) * j((1 + sqrt(-A003173(n+2)))/2).

Original entry on oeis.org

0, 3375, 32768, 884736, 884736000, 147197952000, 262537412640768000

Offset: 1

Seiichi Manyama, Jun 03 2018

			j((1+sqrt(  -3))/2) =                   0.
j((1+sqrt(  -7))/2) =               -3375 = (-1) *     15^3.
j((1+sqrt( -11))/2) =              -32768 = (-1) *     32^3.
j((1+sqrt( -19))/2) =             -884736 = (-1) *     96^3.
j((1+sqrt( -43))/2) =          -884736000 = (-1) *    960^3.
j((1+sqrt( -67))/2) =       -147197952000 = (-1) *   5280^3.
j((1+sqrt(-163))/2) = -262537412640768000 = (-1) * 640320^3.

Eric Weisstein's World of Mathematics, Heegner Number and j-Function

Cf. A003173, A032354, A199743, A305475.

a(n) = A199743(n-1)^3 for n > 1.

A338087 a(n) is the smallest prime number which can be represented as x^2 + h*y^2 with x > 0 and y > 0 for each h in the first n Heegner numbers (A003173).

Original entry on oeis.org

2, 17, 73, 193, 1873, 20353, 20353, 79633, 2333017

Offset: 1

Marco Frigerio, Oct 09 2020

The sequence lists prime numbers, in nondecreasing order, such that each of them can be written, in a unique way, in the form x^2 + h*y^2, where x, y are natural numbers, while h takes an increasing number of values of the sequence A003173 (Heegner numbers). See examples.

			a(1) = 2 because, for A003173(1) = 1, 2 = 1^2+A003173(1)*1^2.
a(2) = 17 because, considered the first two Heegner numbers, A003173(1) = 1 and A003173(2) = 2, 17 = 1^2+A003173(1)*4^2 = 3^2+A003173(2)*2^2.
The prime 20353 is present in the sequence 2 times because:
a(6) = 63^2+A003173(1)*128^2 = 79^2+A003173(2)*84^2 = 55^2+A003173(3)*76^2 = 65^2+A003173(4)*48^2 = 137^2+A003173(5)*12^2 = 97^2+A003173(6)*24^2, with Heegner numbers up to A003173(6)=19, and also:
a(7) = 119^2+A003173(7)*12^2, with Heegner number A003173(7)=43.
2333017 is the last term of the sequence since for every Heegner number h there are x, y such that 2333017 = x^2 + h*y^2 and this is the least prime for which this is possible.
For n=9, h in A003173 = {1,2,3,7,11,19,43,67,163},
   a(9) = 2333017
          = 989^2  +A003173(1)*1164^2
          = 1493^2 +A003173(2)*228^2
          = 1093^2 +A003173(3)*616^2
          = 685^2  +A003173(4)*516^2
          = 1349^2 +A003173(5)*216^2
          = 179^2  +A003173(6)*348^2
          = 1293^2 +A003173(7)*124^2
          = 1395^2 +A003173(8)*76^2
          = 1485^2 +A003173(9)*28^2.

Cf. A003173.

isok(p,u)={for(i=1, #u, my(s=qfbsolve(Qfb(1,0,u[i]),p)); if(s==0 || s[1]==0, return(0))); 1}
a(n)={my(u=[1, 2, 3, 7, 11, 19, 43, 67, 163][1..n]); forprime(p=2, oo, if(isok(p,u), return(p)))}
vector(9, n, a(n)) \\ Andrew Howroyd, Nov 05 2020

A340055 Primes that can be written in the form j^2 + h*k^2, where j and k are positive integers, for every h in A003173 (Heegner numbers).

Original entry on oeis.org

2333017, 5995081, 11414209, 11941273, 12953593, 14823769, 18550849, 19231969, 23582161, 26603977, 27336457, 29236729, 32630161, 35452033, 35836249, 37895089, 40411177, 42911257, 46007329, 46087057, 49680577, 49825609, 52046593, 52208017, 55624297, 63257401

Offset: 1

Marco Frigerio, Dec 29 2020

nonn

The first term in this sequence is equal to last term in A338087.

The sequence is obtained using Lista(m), with m=633*10^5, see section PROG. One can increase m to obtain further terms of the sequence.

			2333017 =  989^2 + A003173(1)*1164^2
        = 1493^2 + A003173(2)*228^2
        = 1093^2 + A003173(3)*616^2
        =  685^2 + A003173(4)*516^2
        = 1349^2 + A003173(5)*216^2
        =  179^2 + A003173(6)*348^2
        = 1293^2 + A003173(7)*124^2
        = 1395^2 + A003173(8)*76^2
        = 1485^2 + A003173(9)*28^2.

Cf. A003173, A338087.

Heegner()={my (d, k, v);  v=vector(3, i, i); for(k=2, 41, d=4*k-1; if(isprime(d) && qfbclassno(-d)==1, v=concat(v, d))); return(v);}
isok(p,u)={my (i, s, n=matsize(u)[2], t=0);for(i=1, n, s=kronecker(-u[i],p); if(s==1, t++,break));if(t==n,t=0;for(i=1, n, s=qfbsolve(Qfb(1,0,u[i]),p); if(s==[], break,t++)));if(t==n,1,0)}
Primo(p, m)={my(u=Heegner()); while(pr,v=concat(v,q),q=m)); return(v);}

A038551 First differences of A003173.

Original entry on oeis.org

1, 1, 4, 4, 8, 24, 24, 96

Offset: 0

Robert Brewer (RBrewerJr(AT)aol.com)

Devlin, Keith, Mathematics: The New Golden Age.

A118563 The set of numbers of the form A003173(n)+1 with odd A003173(n) or of the form 2*(A003173(n)+1) with even A003173(n).

Original entry on oeis.org

2, 4, 6, 8, 12, 20, 44, 68, 164

Offset: 1

Roger L. Bagula, May 03 2006

Sorted sequence of numbers of the form A109043(A003173(k)), any k.

Cf. A003173.

h = {1, 2, 3, 7, 11, 19, 43, 67, 163} a = Union[Table[If[Mod[h[[n]] + 1, 2] == 0, h[[n]] + 1, 2*(h[[n]] + 1)], {n, 1, Length[h]}]]

keywords fini,full,less added by Assoc. Eds. of the OEIS, Jun 15 2010

A151992 Erroneous version of A003173.

Original entry on oeis.org

1, 2, 4, 7, 11, 19, 43, 67, 163

Offset: 1

dead

Seen on the Web. Included here to serve as a pointer to the correct version.

A340133 The sequence lists the least prime numbers, in ascending order, such that each of them can be written, in a unique way, in the form x^2 + h*y^2, where x, y are natural numbers, while h takes all the values of the sequences A000926 (Idoneal numbers) and A003173 (Heegner numbers). See example.

Original entry on oeis.org

3230498881, 5086789009, 6956459689, 7260636769, 12387462649, 13125124321, 14049841129, 14247509329, 14310889849, 15871864849, 16573389361, 17502040609, 17768627809, 22042168201, 22621870441, 22957650769, 23018043409, 23819076121, 25228204849, 26585136601

Offset: 1

Marco Frigerio, Dec 29 2020

nonn

First number in this sequence is equal to least common number of sequences A340055 and A340132.

The sequence is obtained using Lista(m), with m=266*10^8, see section PROG. It's possible increase m to discover more terms of the sequence. It's also possible to extend the sequences A340055 and A340132 to check their common numbers.

			3230498881 = 2465^2+A000926(1)*56784^2
           = 56609^2+A000926(2)*3600^2
           = 35927^2+A000926(3)*25428^2
           = ...
           = 56791^2+A003173(9)*180^2
           = ...
           = 35743^2+A000926(65)*1028^2

Cf. A000926, A003173, A340055, A340132.

Union()={ my (v);v=(select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848]));for(k=3, 41, d=4*k-1; if(isprime(d) && qfbclassno(-d)==1, v=concat(v, d)));return(v);}
isok(p,u)={my (i, s, n=matsize(u)[2], t=0);for(i=1, n, s=kronecker(-u[i],p); if(s==1, t++,break));if(t==n,t=0;for(i=1, n, s=qfbsolve(Qfb(1,0,u[i]),p); if(s==[], break,t++)));if(t==n,1,0)}
Primo(p, m)={my(u=Union()); while(pr,v=concat(v,q),q=m)); return(v);}

cp's OEIS Frontend

A199743 Rounded near-integers (exp(Pi*sqrt(h)) - 744)^(1/3) where h is A003173(n+3) (Heegner numbers of the form 4p-1 where p is prime).

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A242216 Number of partitions of n into Heegner numbers, cf. A003173.

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A242217 Number of partitions of n into distinct Heegner numbers, cf. A003173.

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A305500 a(n) = (-1) * j((1 + sqrt(-A003173(n+2)))/2).

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A338087 a(n) is the smallest prime number which can be represented as x^2 + h*y^2 with x > 0 and y > 0 for each h in the first n Heegner numbers (A003173).

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A340055 Primes that can be written in the form j^2 + h*k^2, where j and k are positive integers, for every h in A003173 (Heegner numbers).

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A038551 First differences of A003173.

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A118563 The set of numbers of the form A003173(n)+1 with odd A003173(n) or of the form 2*(A003173(n)+1) with even A003173(n).

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A151992 Erroneous version of A003173.

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