A000926 -id:A000926 - cp's OEIS frontend

A034168 Disjoint discriminants (one form per genus) of type 2 (doubled).

2, 6, 10, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset: 1

Jonathan Borwein (jborwein(AT)cecm.sfu.ca), choi(AT)cecm.sfu.ca (Stephen Choi)

J. M. Borwein and P. B. Borwein, Pi and the AGM, page 293.
L. E. Dickson, Introduction to the theory of numbers, Dover, NY, 1929.

J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.
Experimental Mathematics, Home Page

Cf. A000926, A005843, A034169, A055745, A139826. Subsequence of A025052.

noSol = {};
Do[lim = Ceiling[(n-2)/3]; found = False; Do[If[n > a*b && Mod[n - a*b, a+b] == 0 && Quotient[n - a*b, a+b] > b, found = True; Break[]], {a, 1, lim-1}, {b, a+1, lim}]; If[!found, AppendTo[noSol, n]], {n, 1000}];
Select[noSol, EvenQ[#] && SquareFreeQ[#]&] (* Jean-François Alcover, Jul 21 2022, after T. D. Noe in A000926 *)

ok(n)={n%4==2 && issquarefree(n) && !select(t->t<>2, quadclassunit(-4*n).cyc)} \\ Andrew Howroyd, Jun 09 2018

Intersection of A005843 and A139826. - Andrew Howroyd, Jun 09 2018

A132213 Number of distinct primes among the squares mod n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 0, 0, 2, 2, 4, 1, 1, 3, 3, 0, 2, 4, 3, 0, 4, 1, 4, 1, 2, 4, 2, 1, 3, 6, 2, 0, 5, 2, 6, 2, 2, 7, 5, 0, 6, 5, 3, 3, 8, 6, 3, 0, 3, 6, 8, 0, 6, 8, 3, 2, 2, 3, 7, 3, 3, 2, 7, 0, 9, 10, 3, 4, 6, 4, 9, 1, 10, 10, 11, 1, 2, 13, 3, 0, 10, 4, 5, 4, 4, 13, 4, 1, 11, 10, 4, 4

Offset: 1

T. D. Noe, Aug 13 2007, Aug 17 2007

It appears that a(n)=0 for only the 30 numbers in A065428, which appears to be related to idoneal numbers, A000926. The graph shows a(n) can be quite small even for large n. For example, a(9240)=7. Observe that the graph up to n=10000 appears to have 5 components. Why?
The logarithmic plot of the first 10^6 terms shows seven components.
From Rémy Sigrist, Nov 28 2017: (Start)
Empirically, in the logarithmic plot of the sequence:
- the set of indices of the first component (starting from the top), say S_1, is the union of A061345 and of A278568,
- the set of indices of the n-th component (for n > 1), say S_n, contains the numbers k not in a previous component and such that (omega(k) = n-1) or (omega(k) = n and val(k) = 0 or 2) or (omega(k) = n+1 and val(k) = 1) (where omega(k) = A001221(k) and val(k) = A007814(k)),
- see logarithmic scatterplot colored according to this scheme in Links section.
(End)

			For n=14, the squares (mod n) repeat 0,1,4,9,2,11,8,7,8,11,2,9,4,1,0,..., a sequence containing three distinct primes: 2, 7 and 11. Hence a(14)=3.

T. D. Noe, Table of n, a(n) for n = 1..10000
T. D. Noe, Logarithmic plot of 10^6 terms
Rémy Sigrist, Colored logarithmic plot of 2*10^6 terms

Cf. A000224 (number of squares mod n).

Cf. A000290, A001221, A007814, A010051, A061345, A278568.

import Data.List (nub, genericTake)
a132213 n = sum $ map a010051' $
            nub $ genericTake n $ map (`mod` n) $ tail a000290_list
-- Reinhard Zumkeller, Jun 23 2015, Oct 15 2011

Table[s=Union[Mod[Range[n]^2,n]]; Length[Select[s,PrimeQ]], {n,10000}]
Table[Count[Union[PowerMod[Range[n],2,n]],?PrimeQ],{n,100}] (* _Harvey P. Dale, Mar 02 2018 *)

A229461 Numbers k such that there is no convex pentagon that can be decomposed into k pairwise congruent regular equilateral triangles.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 16, 18, 21, 22, 24, 25, 30, 33, 37, 40, 42, 45, 48, 57, 58, 70, 72, 78, 85, 88, 93, 102, 105, 120, 130, 133, 165, 168, 177, 190, 210, 232, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848

Offset: 1

Suggested by Eike Hertel, Hugo Pfoertner, Sep 24 2013

nonn

Conjecture: These 59 numbers are all such exceptions.
Terms are idoneal numbers (A000926) except for the six terms of A229462.
Numbers k not expressible as k = x^2 - y^2 - z^2 with x,y,z >= 1 and x > y+z.

Eike Hertel, Reguläre Dreieckspflasterungen konvexer Polygone, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/01/13, 2013 (preprint).
Eike Hertel and Christian Richter, Tiling Convex Polygons with Congruent Equilateral Triangles, Discrete Comput Geom (2014) 51:753-759.
E. Kani, Idoneal numbers and some generalizations, Ann Sci. Math. Québec, 35 (2011), pp. 197-227.
Kival Ngaokrajang, Illustration of initial terms

Cf. A000926 (idoneal numbers), A229462 (idoneal numbers not in this sequence), A229757 (hexagon exception numbers), A025052 (numbers not of form a*b+b*c+c*a).

A232529 Least positive integer m such that for all primes p where p and p-n are quadratic residues (mod 4n), (m^2)p can be written as x^2+n*y^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 1, 3, 1, 4, 2, 1, 5, 2, 2, 1, 3, 4, 1, 9, 1, 2, 3, 1, 5, 8, 1, 5, 3, 2, 2, 3, 3, 4, 3, 1, 1, 6, 1, 5, 9, 3, 2, 3, 5, 2, 6, 5, 1, 12, 1, 7, 9, 2, 4, 3, 1, 8, 3, 3, 7, 6, 2, 1, 9, 4, 1, 15, 3, 2, 3, 1, 25, 8, 2, 7, 7, 2, 2, 15

Offset: 1

V. Raman, Nov 25 2013

If n is a convenient number (A000926), then a(n) = 1.

m is also the lowest nonzero integer such that m^2 can be generated by using all the inequivalent primitive quadratic forms of discriminant = -4n.

			For n = 59, all primes p such that p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n) are either of the form x^2+59*y^2 or 4*x^2+2*x*y+15*y^2 or 3*x^2+2*x*y+20*y^2 or 5*x^2+2*x*y+12*y^2 or 7*x^2+4*x*y+9*y^2.
We have (6^2)*(x^2+59*y^2) = (6*x)^2+59*(6*y)^2,
(6^2)*(4*x^2+2*x*y+15*y^2) = (12*x+3*y)^2 + 59*(3*y)^2,
(6^2)*(7*x^2+4*x*y+9*y^2) = (4*x+18*y)^2 + 59*(2*x)^2,
(6^2)*(3*x^2+2*x*y+20*y^2) = (7*x+22*y)^2 + 59*(x-2*y)^2,
(6^2)*(5*x^2+2*x*y+12*y^2) = (11*x+14*y)^2 + 59*(x-2*y)^2.
So, m = 6 satisfies this condition for n = 59: for all primes p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n), (m^2)*p can be written as x^2+n*y^2.
And m = 6 is the smallest value of m to satisfy this condition. So, a(59) = 6.

Cf. A232530, A000926.

a(n)=sqrt(A232530(n)).

A232530 Least square m^2 such that for all primes p where p and p-n are quadratic residues (mod 4n), (m^2)p can be written as x^2+n*y^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 1, 1, 9, 1, 4, 4, 1, 1, 9, 1, 1, 9, 4, 1, 9, 1, 16, 4, 1, 25, 4, 4, 1, 9, 16, 1, 81, 1, 4, 9, 1, 25, 64, 1, 25, 9, 4, 4, 9, 9, 16, 9, 1, 1, 36, 1, 25, 81, 9, 4, 9, 25, 4, 36, 25, 1, 144, 1, 49, 81, 4, 16, 9, 1, 64, 9, 9, 49, 36, 4, 1, 81, 16, 1, 225, 9, 4, 9, 1, 625, 64, 4, 49

Offset: 1

V. Raman, Nov 25 2013

If n is a convenient number (A000926) then a(n) = 1.

m^2 is also the smallest positive square that can be generated by using all the inequivalent primitive quadratic forms of discriminant = -4n.

			For n = 11, all primes p such that p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n) are either of the form x^2+11*y^2 or 3*x^2+2*x*y+4*y^2.
4*(x^2+11*y^2) = (2*x)^2+11*(2*y)^2, 4*(3*x^2+2*x*y+4*y^2) = (x+4*y)^2+11*x^2. Also, 4 is the smallest square to satisfy this condition. So, a(11) = 4.
For n = 14, all primes p such that p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n) are either of the form x^2+14*y^2 or 2*x^2+7*y^2.
9*(x^2+14*y^2) = (3*x)^2+14*(3*y)^2, 9*(2*x^2+7*y^2) = (2*x+7*y)^2+14*(x-y)^2 = (2*x-7*y)^2+14*(x+y)^2. Also, 9 is the smallest square to satisfy this condition. So, a(14) = 9.
For n = 17, all primes p such that p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n) are either of the form x^2+17*y^2 or 2*x^2+2*x*y+9*y^2.
9*(x^2+17*y^2) = (3*x)^2+17*(3*y)^2, 9*(2*x^2+2*x*y+9*y^2) = (x+9*y)^2+17*x^2. Also, 9 is the smallest square to satisfy this condition. So, a(17) = 9.
For n = 19, all primes p such that p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n) are either of the form x^2+19*y^2 or 4*x^2+2*x*y+5*y^2.
4*(x^2+19*y^2) = (2*x)^2+19*(2*y)^2, 4*(4*x^2+2*x*y+5*y^2) = (4*x+y)^2+19*y^2. Also, 4 is the smallest square to satisfy this condition. So, a(19) = 4.
For n = 20, all primes p such that p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n) are either of the form x^2+20*y^2 or 4*x^2+5*y^2.
4*(x^2+20*y^2) = (2*x)^2+20*(2*y)^2, 4*(4*x^2+5*y^2) = (4*x)^2+20*y^2. Also, 4 is the smallest square to satisfy this condition. So, a(20) = 4.

Cf. A232529, A000926.

a(n)=A232529(n)^2.

A230391 Numbers m such that 232*m^2+1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 12, 13, 15, 16, 17, 18, 20, 22, 24, 26, 27, 28, 31, 35, 36, 37, 38, 44, 45, 46, 48, 49, 50, 52, 53, 62, 67, 71, 72, 73, 74, 76, 79, 81, 82, 86, 87, 94, 95, 99, 100, 104, 106, 107, 112, 113, 115, 118, 119, 121, 124, 126, 127, 136, 138

Offset: 1

Bruno Berselli, Oct 18 2013

The form "232aa + 1" has been used by Euler to find idoneal numbers (A000926), and 232 itself is an idoneal number (see References).

Numbers m for which 232*m^2+1 is not prime are: 0, 4, 8, 11, 14, 19, 21, 23, 25, 29, 30, 32, 33, 34, 39, 40, 41, 42, 43, 47, ... (see table on page 14 of Euler's paper).

Leonhard Euler, Facillima methodus plurimos numeros primos praemagnos inveniendi, Nova Acta Academiae Scientiarum Imperialis Petropolitanae Tomus XIV (1805), Mathematica et Physico-Mathematica (this sequence is on page 10).

Bruno Berselli, Table of n, a(n) for n = 1..1000
Umberto Cerruti, I numeri idonei di Eulero (in Italian), p. 3.
Leonhard Euler, An easy method for finding many very large prime numbers, p. 8, arXiv:math/0507401 [math.HO], 2005-2008. Translated from Latin.

Cf. A000926, A230392 (associated primes).

[n: n in [1..200] | IsPrime(232*n^2+1)];

Select[Range[200], PrimeQ[232 #^2 + 1] &]

is(n)=isprime(232*n^2+1) \\ Charles R Greathouse IV, Jun 06 2017

A230392 Primes of the form 232*m^2+1.

Original entry on oeis.org

233, 929, 2089, 5801, 8353, 11369, 18793, 23201, 33409, 39209, 52201, 59393, 67049, 75169, 92801, 112289, 133633, 156833, 169129, 181889, 222953, 284201, 300673, 317609, 335009, 449153, 469801, 490913, 534529, 557033, 580001, 627329, 651689, 891809, 1041449

Offset: 1

Bruno Berselli, Oct 18 2013

nonn

Nonprime numbers of this form are: 1, 3713, 14849, 28073, 45473, 83753, 102313, 122729, 145001, 195113, 208801, 237569, 252649, 268193, ...

Leonhard Euler, Facillima methodus plurimos numeros primos praemagnos inveniendi, Nova Acta Academiae Scientiarum Imperialis Petropolitanae Tomus XIV (1805), Mathematica et Physico-Mathematica.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Umberto Cerruti, I numeri idonei di Eulero (in Italian), p. 4.
Euler Archive, E718 -- Facillima methodus plurimos numeros primos praemagnos inveniendi
Leonhard Euler, An easy method for finding many very large prime numbers, arXiv:math/0507401 [math.HO], 2005-2008. Translated from Latin.

Cf. A000926, A230391 (associated n).

[m: n in [1..100] | IsPrime(m) where m is 232*n^2+1];

Select[Table[232 n^2 + 1, {n, 100}], PrimeQ]

A232528 Numbers n such that for all primes p where p and p-n are quadratic residues (mod 4n), 4p can be written as x^2 + n*y^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 37, 40, 42, 43, 45, 48, 51, 52, 57, 58, 60, 64, 67, 70, 72, 75, 78, 84, 85, 88, 91, 93, 96, 99, 100, 102, 105, 112, 115, 120, 123, 130, 132, 133, 147, 148, 160, 163, 165, 168, 177, 180, 187, 190, 192, 195

Offset: 1

V. Raman, Nov 25 2013

nonn

Convenient numbers (A000926) are numbers n such that for all primes p where p and p-n are quadratic residues (mod 4*n), p can be written as x^2 + n*y^2, so convenient numbers are a subsequence.
All non-convenient numbers which are members of this sequence are either congruent to 0 (mod 4) or 3 (mod 4).
The equation 4*p = x^2 + n*y^2 is important because if n is a squarefree integer congruent to 3 (mod 4), then the ring of integers Q[sqrt(-n)] will be all integers of form (x/2) + (y/2)*sqrt(-n) for x and y of the same parity, whose norm is (x/2)^2 + n*(y/2)^2. If prime p = (x/2)^2 + n*(y/2)^2, then 4*p = x^2 + n*y^2.
Is this sequence finite?
Is 7392 the largest term of this sequence?
There are no further terms up to 10^6. - Andrew Howroyd, Jun 08 2018

			n = 14 is not a member of this sequence because for prime p = 71, 4*p = 284 cannot be written as x^2 + 14*y^2.

V. Raman, Table of n, a(n) for n = 1..139
Thomas R. Hagedorn, Primes of the form x^2+ny^2 and the geometry of (convenient) numbers

Cf. A000926.

ok(n)=!#select(k->k<>2, quadclassunit(-n*if((-n)%4>1, 4, 1)).cyc) \\ Andrew Howroyd, Jun 08 2018

A234287 Number of distinct quadratic forms of discriminant -4n by which some prime can be represented.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 2, 3, 5, 3, 4, 4, 3, 4, 4, 3, 4, 4, 3, 5, 5, 2, 4, 4, 5, 5, 4, 3, 5, 5, 3, 4, 5, 4, 5, 5, 4, 4, 5, 4, 7, 4, 2, 6, 5, 4, 5, 5, 4, 6, 6, 3, 6, 6, 4, 5, 6, 3, 6, 6, 5, 6, 4, 4, 7, 5, 3, 6, 7, 4, 6, 5, 5, 7, 7, 5, 5, 4, 5, 6, 7, 3, 6, 6, 5

Offset: 1

V. Raman, Dec 22 2013

nonn

This is similar to A232551, except that this includes non-primitive quadratic forms like 2x^2+2xy+4y^2 and 2x^2+4y^2 because the prime 2 can be represented by both of them. But unlike A067752, we do not include quadratic forms like 4x^2+2xy+4y^2 and 4x^2+4xy+4y^2 by which no prime can be represented.
So, when n == 3 (mod 4), this includes the additional non-primitive quadratic form 2x^2+2xy+((n+1)/2)y^2 and when p^2 divides n, where p is prime, this includes the additional non-primitive quadratic form px^2+(n/p)y^2.
If p is a prime and if p^2 does not divide n, then there exist a unique non-primitive quadratic form of discriminant = -4n by which p can be represented if and only if -n is a quadratic residue (mod p) and there exists a multiple of p which can be written in the form x^2+ny^2 in which p appears raised to an odd power, except when p = 2 and n == 3 (mod 8).

V. Raman, Examples of these distinct quadratic forms for n = 1..100

Cf. A000003, A000926, A067752, A232550, A232551.

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

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 A000926 (idoneal numbers). See examples.

			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.

Marco Frigerio, Table of n, a(n) for n = 1..65

Cf. A000926.

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

cp's OEIS Frontend

A034168 Disjoint discriminants (one form per genus) of type 2 (doubled).

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A132213 Number of distinct primes among the squares mod n.

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A229461 Numbers k such that there is no convex pentagon that can be decomposed into k pairwise congruent regular equilateral triangles.

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A232529 Least positive integer m such that for all primes p where p and p-n are quadratic residues (mod 4*n), (m^2)*p can be written as x^2+n*y^2.

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A232530 Least square m^2 such that for all primes p where p and p-n are quadratic residues (mod 4*n), (m^2)*p can be written as x^2+n*y^2.

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A230391 Numbers m such that 232*m^2+1 is prime.

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Magma

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A230392 Primes of the form 232*m^2+1.

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A232528 Numbers n such that for all primes p where p and p-n are quadratic residues (mod 4*n), 4*p can be written as x^2 + n*y^2.

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A234287 Number of distinct quadratic forms of discriminant -4n by which some prime can be represented.

A232529 Least positive integer m such that for all primes p where p and p-n are quadratic residues (mod 4n), (m^2)p can be written as x^2+n*y^2.

A232530 Least square m^2 such that for all primes p where p and p-n are quadratic residues (mod 4n), (m^2)p can be written as x^2+n*y^2.

A232528 Numbers n such that for all primes p where p and p-n are quadratic residues (mod 4n), 4p can be written as x^2 + n*y^2.