Marco Frigerio has authored 5 sequences.
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
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
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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);}
A340132
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 sequence A000926 (idoneal numbers).
Original entry on oeis.org
1083289, 3818929, 6104641, 6868801, 7623529, 8465209, 9033649, 10105489, 11400481, 11597569, 11809561, 12338041, 12348961, 13154761, 13426009, 15861169, 16889161, 16922161, 18596449, 19684729, 20322481, 21067201, 21480001, 22684561, 23654569, 24531049
Offset: 1
1083289 = 315^2 + A000926(1)*992^2
= 1033^2 + A000926(2)*90^2
= 979^2 + A000926(3)*204^2
= ...
= 817^2 + A000926(65)*15^2.
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Idoneal()={return(select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848]));}
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=Idoneal()); while(pr,v=concat(v,q),q=m)); return(v);}
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
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.
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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);}
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
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.
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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
A338088
Smallest prime numbers which can be represented as x^2 + h*y^2 with x > 0 for every h in the first n idoneal numbers.
Original entry on oeis.org
2, 17, 73, 73, 241, 241, 1009, 1009, 1009, 1009, 1009, 2521, 2521, 2521, 2521, 2521, 8089, 8089, 8089, 8089, 8089, 8089, 19009, 19009, 19009, 19009, 19009, 19009, 53881, 53881, 53881, 53881, 53881, 53881, 53881, 605641, 605641, 605641, 605641, 605641, 605641
Offset: 1
a(1) = 2 because, for A000926(1) = 1, 2 = 1^2+A000926(1)*1^2.
a(2) = 17 because, considered the first two idoneal numbers, A000926(1) = 1 and A000926(2) = 2, 17 = 1^2+A000926(1)*4^2 = 3^2+A000926(2)*2^2.
The prime 1009 is present in the sequence 5 times because:
a(7) = 15^2+1*28^2 = 19^2+2*18^2 = 31^2+3*4^2 = 15^2+4*14^2 = 17^2+5*12^2 = 25^2+6*12^2 = 1^2+7*12^2, with idoneal numbers up to A000926(7), and also:
a(8) = 19^2+8*9^2,
a(9) = 28^2+9*5^2,
a(10) = 3^2+10*10^2,
a(11) = 31^2+12*2^2,
with idoneal numbers from A000926(8) to A000926(11).
1083289 is the last term of the sequence since for every idoneal number h there are x, y such that 1083289 = x^2 + h*y^2 and this is the least prime for which this is possible.
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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}
idoneal()={select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848])}
seq()={my(u=idoneal(), v=[1], L=List()); forprime(p=2, oo, if(isok(p,v), listput(L,p); my(k=#v); while(k<#u && isok(p,[u[k+1]]), listput(L,p); k++); if(k==#u, return(Vec(L))); v=u[1..k+1]))} \\ Andrew Howroyd, Nov 05 2020
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