A025052 -id:A025052 - cp's OEIS frontend

A000926 Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848

Offset: 1

N. J. A. Sloane

There are many equivalent definitions of these numbers. Based on Cox, Theorem 3.22 and Proposition 3.24 and a comment by Eric Rains (rains(AT)caltech.edu), we can say that a positive number n belongs to this sequence if and only if any of the following equivalent statements is true:
(1) Let m > 1 be an odd number relatively prime to n which can be written in the form x^2 + n*y^2 with x, y relatively prime. If the equation m = x^2 + n*y^2 has only one solution with x, y >= 0, then m is a prime number. [Euler]
(2) Every genus of quadratic forms of discriminant -4n consists of a single class. [Gauss]
(3) If a*x^2 + b*x*y + c*y^2 is a reduced quadratic form of discriminant -4n, then either b=0, a=b or a = c. [Cox]
(4) Two quadratic forms of discriminant -4n are equivalent if and only if they are properly equivalent. [Cox]
(5) The class group C(-4n) is isomorphic to (Z/2Z)^m for some integer m. [Cox]
(6) n is not of the form ab+ac+bc with 0 < a < b < c. (See proof in link below.)  [Rains]
It is conjectured that the list given here is complete. Chowla showed that the list is finite and Weinberger showed that there is at most one further term.
If an additional term exists it is > 100000000. - Jud McCranie, Jun 27 2005
The terms shown are the union of {1,2,3,4,7}, A033266, A033267, A033268 and A033269 (corresponding to class numbers 1, 2, 4, 8 and 16 respectively).
Note that for n in this sequence, n+1 is either a prime, twice a prime, the square of a prime, 8 or 16. - T. D. Noe, Apr 08 2004. [This is a general theorem that is not hard to prove using genus theory. The "32" in the original comment was an error. - Tom Hagedorn (hagedorn(AT)tcnj.edu), Dec 29 2008]
Also numbers n such that for all primes p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n), p can be uniquely written into the form as x^2+n*y^2. - V. Raman, Nov 25 2013

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 97 at p. 272.
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Section 3.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1848, p. 146, Ellipses, Paris 2008.
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, Sections 329-334.
G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.
Paulo Ribenboim, My Numbers, My Friends, Chapter 11, Springer-Verlag, NY, 2000.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 142-143.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 103.
A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see pp. 188, 219-226.

S. Chowla, An extension of Heilbronn's class number theorem, Quart. J. math., 5 (1934), 304-307.
K. S. Brown, Mathpages, Numeri Idonei
Günther Frei, Les nombres convenables de Leonhard Euler, Publications Université de Besançon, 1983-1984.
Günther Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55-58 and 64.
E. Hertel, C. Richter, Tiling Convex Polygons with Congruent Equilateral Triangles, Discrete & Computational Geometry, 2014, DOI 10.1007/s00454-014-9576-7. Mentions this sequence. - _N. J. A. Sloane_, Mar 17 2014
O.-H. Keller, Über die "Numeri idonei" von Euler, Beitraege Algebra Geom., 16 (1983), 79-91. [Math. Rev. 85m:11019]
Robert Krzyzanowski, Euler's Convenient Numbers
David Masser, Alan Baker, arXiv:2010.10256 [math.HO], 2020. See p. 24.
Eric Rains, Comments on A000926
Paulo Ribenboim, Galimatias Arithmeticae, in Mathematics Magazine 71(5) 339 1998 MAA.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. Steinig, On Euler's ideoneal numbers, Elemente Math., 21 (1966), 73-88.
M. Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004
P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117-124.
Eric Weisstein's World of Mathematics, Idoneal Number

Sequence A025052 is a subsequence.
Cf. A014556, A026501, A093669, A094376, A094377, A094378.
Cf. A139642 (congruences for idoneal quadratic forms).

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, 10000}]; noSol (* T. D. Noe, Apr 08 2004 *)

A000926(Nmax=1e9)={for(n=1,Nmax,for(a=1,sqrtint(n\3),for(b=a+1,(n-a)\(3*a+2),n-a<(2*a+1+b)*b & break;(n-a*b)%(a+b)==0 & next(3)));print1(n", "))} \\ M. F. Hasler, Dec 04 2007

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

Edited by N. J. A. Sloane, Dec 07 2007

A093669 Numbers having a unique representation as ab+ac+bc, with 0 < a < b < c.

Original entry on oeis.org

11, 14, 17, 19, 20, 27, 32, 34, 36, 43, 46, 49, 52, 64, 67, 73, 82, 97, 100, 142, 148, 163, 193

Offset: 1

T. D. Noe, Apr 08 2004

Are there more terms?

No more terms < 10^6. - Joerg Arndt, Oct 01 2017

			11 is on the list because 11 = 1*2+1*3+2*3.

See A025052.

Cf. A025052, A025058, A025060.

Cf. A000926 (numbers not of the form ab+ac+bc, 0

oneSol={}; Do[lim=Ceiling[(n-2)/3]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, cnt++; If[cnt>1, Break[]]], {a, 1, lim-1}, {b, a+1, lim}]; If[cnt==1, AppendTo[oneSol, n]], {n, 10000}]; oneSol

from collections import Counter
def aupto(N):
    acount = Counter()
    for i in range(1, N-1):
        for j in range(i+1, N//i + 1):
            p, s = i*j, i+j
            for k in range(j+1, (N-p)//s + 1):
                acount.update([p + s*k])
    return sorted([k for k in acount if acount[k] == 1])
print(aupto(10**5)) # Michael S. Branicky, Nov 14 2021

A025051 Numbers of the form jk + ki + i*j, where i,j,k >= 1.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Clark Kimberling

J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Uniquification of sequence A025050.

Cf. A025052 (complement), A066955.

Take[Union[Total[Times@@@Subsets[#,{2}]]&/@Tuples[Range[25],{3}]], 80] (* Harvey P. Dale, Aug 11 2011 *)

On the GRH, a(n) = n + 18 for n > 444. If there is some N > 444 with a(N) not equal to n + 19, then for n >= N, a(n) = n + 19 (and GRH is false). See Borwein & Choi. - Charles R Greathouse IV, Dec 19 2022

n is in the sequence if and only if A066955(n) > 0. - Charles R Greathouse IV, Jun 25 2024

A027563 Numbers not of form abc + abd + acd + bcd for 1<=a<=b<=c<=d.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 21, 23, 26, 29, 30, 33, 35, 41, 48, 51, 53, 63, 65, 74, 75, 86, 89, 90, 98, 111, 113, 119, 125, 131, 141, 155, 158, 173, 179, 191, 209, 210, 233, 239, 251, 254, 273, 285, 293, 321, 323, 326, 338, 341, 345, 363, 419

Offset: 1

R. H. Hardin

This list is conjecturally complete, but this has not been proved. It may be complete as a consequence of the Generalized Riemann Hypothesis; see comments for A025052. - Harry Richman, Jan 09 2025

T. D. Noe, Table of n, a(n) for n=1..126 (complete sequence)

Cf. A025052 (3 variables), A027564 (5 variables), A027565, A027566.

A094376 Least number having exactly n representations as ab+ac+bc with 0 < a < b < c.

Original entry on oeis.org

1, 11, 23, 41, 47, 59, 71, 116, 119, 131, 164, 425, 191, 236, 239, 446, 335, 419, 311, 404, 431, 584, 647, 524, 479, 1019, 831, 776, 671, 944, 719, 1076, 839, 1004, 959, 1889, 1196, 2099, 1271, 1856, 1151, 1931, 1391, 1676, 1319, 1616, 1751, 3275, 1511

Offset: 0

T. D. Noe and Robert G. Wilson v, Apr 28 2004

nonn

Note that the Mathematica program computes A094376, A094377 and A094378, but outputs only this sequence.

			a(2) = 23 because 23 is the least number with 2 representations: (a,b,c) = (1,2,7) and (1,3,5).

See A025052

Robert Israel, Table of n, a(n) for n = 0..826

Cf. A000926 (n having no representations), A093669 (n having one representation), A025052, A094377, A094378.

f:= proc(n) local a, t, s;
  t:= 0;
  for a from 1 to floor(sqrt(n/3)) do
    t:= t + nops(select(s -> s > 2*a and n+a^2 > s^2, numtheory:-divisors(n+a^2)))
  od;
  t
end proc:
N:= 200: # for a(0)..a(N)
V:= Array(0..N): count:= 0:
for n from 1 while count < N+1 do
   v:= f(n); if v <= N and V[v] = 0 then
      count:= count+1; V[v]:= n; fi
od:
seq(V[i],i=0..N); # Robert Israel, May 05 2021

cntMax=10; nSol=Table[{0, 0, 0}, {cntMax+1}]; Do[lim=Ceiling[(n-2)/3]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, cnt++; If[cnt>cntMax, Break[]]], {a, 1, lim-1}, {b, a+1, lim}]; If[cnt<=cntMax, If[nSol[[cnt+1, 1]]==0, nSol[[cnt+1, 1]]=n]; nSol[[cnt+1, 2]]=n; nSol[[cnt+1, 3]]++;], {n, 10000}]; Table[nSol[[i, 1]], {i, cntMax+1}]

A094377 Greatest number having exactly n representations as ab+ac+bc with 0 < a < b < c.

Original entry on oeis.org

1848, 193, 1012, 862, 3040, 2062, 4048, 3217, 7392, 4162, 7837, 8002, 12397, 13297, 14722, 16417, 21253, 21058, 30493, 27358, 34357, 34318, 47338, 40177, 50317, 39502, 61462, 62302, 73117, 83218, 106177, 67138, 92698, 102958, 134773, 111577, 112942, 121522, 104938, 96958, 151237, 166798, 150382, 139393, 190513, 129838

Offset: 0

T. D. Noe, Apr 28 2004

nonn

Numbers up to 250,000 were checked. Note that the Mathematica program computes A094376, A094377 and A094378, but outputs only this sequence.

			a(1) = 193 because 193 is the largest number with a unique representation: (a,b,c) = (4,7,15).

See A025052.

Cf. A000926 (n having no representations), A093669 (n having one representation), A094376, A094378.

cntMax=10; nSol=Table[{0, 0, 0}, {cntMax+1}]; Do[lim=Ceiling[(n-2)/3]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, cnt++; If[cnt>cntMax, Break[]]], {a, 1, lim-1}, {b, a+1, lim}]; If[cnt<=cntMax, If[nSol[[cnt+1, 1]]==0, nSol[[cnt+1, 1]]=n]; nSol[[cnt+1, 2]]=n; nSol[[cnt+1, 3]]++;], {n, 10000}]; Table[nSol[[i, 2]], {i, cntMax+1}]

More terms (using limit 10^6) from Joerg Arndt, Oct 01 2017

A094379 Least number having exactly n representations as ab+ac+bc with 1 <= a <= b <= c.

Original entry on oeis.org

1, 3, 11, 23, 35, 47, 59, 71, 95, 188, 119, 164, 231, 191, 215, 239, 299, 356, 335, 311, 404, 431, 591, 584, 524, 479, 551, 656, 831, 776, 671, 719, 791, 839, 1004, 1031, 959, 1244, 1196, 1439, 1271, 1151, 1931, 1847, 1391, 1319, 1811, 1784, 1616, 1511, 1799

Offset: 0

T. D. Noe, Apr 28 2004

nonn

Note that the Mathematica program computes A094379, A094380 and A094381, but outputs only this sequence.

A066955(a(n)) = n and A066955(m) = n for m < a(n). [Reinhard Zumkeller, Mar 23 2012]

			a(3) = 23 because 23 is the least number with 3 representations: (a,b,c) = (1,1,11), (1,2,7) and (1,3,5).

See A025052

Reinhard Zumkeller, Table of n, a(n) for n = 0..125

Cf. A025052 (n having no representations), A093670 (n having one representation), A094380, A094381.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a094379 = (+ 1) . fromJust . (`elemIndex` a066955_list)
-- Reinhard Zumkeller, Mar 23 2012

cntMax=10; nSol=Table[{0, 0, 0}, {cntMax+1}]; Do[lim=Ceiling[(n-1)/2]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>=b, cnt++; If[cnt>cntMax, Break[]]], {a, 1, lim}, {b, a, lim}]; If[cnt<=cntMax, If[nSol[[cnt+1, 1]]==0, nSol[[cnt+1, 1]]=n]; nSol[[cnt+1, 2]]=n; nSol[[cnt+1, 3]]++;], {n, 10000}]; Table[nSol[[i, 1]], {i, cntMax+1}]

A027564 Numbers not of form abcd + abce + abde + acde + bcde for 1 <= a <= b <= c <= d <= e.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 18, 19, 20, 22, 24, 26, 27, 31, 32, 34, 35, 36, 38, 39, 42, 46, 47, 50, 54, 55, 59, 60, 62, 66, 67, 70, 71, 75, 78, 84, 87, 90, 92, 94, 98, 99, 102, 104, 106, 110, 111, 115, 116, 119, 122, 126, 127, 130, 131, 132, 138

Offset: 1

R. H. Hardin

Robert Israel, Table of n, a(n) for n = 1..1578

Cf. A025052 (3 variables), A027563 (4 variables), A027565, A027566.

N := 1000: # for all terms <= N
V:= Vector(N):
for a from 1 to floor((N/5)^(1/4)) do
  for b from a while 4*a*b^3+b^4<= N do
    for c from b while 3*a*b*c^2 + (a+b)*c^3 <= N do
      for d from c while 2*a*b*c*d + (b*c+a*c+a*b)*d^2 <= N do
        for e from d do
          r:= a*b*c*d+a*b*c*e+a*b*d*e+a*c*d*e+b*c*d*e;
          if r > N then break fi;
          V[r]:= 1;
od od od od od:
select(t -> V[t]=0, [$1..N]); # Robert Israel, Nov 04 2018

A027565 Largest number not of form k_1 k_2 .. k_n (1/k_1 + .. + 1/k_n), k_i integers >= 1.

Original entry on oeis.org

1, 462, 52061, 33952307

Offset: 2

R. H. Hardin

			For n=2, the largest number not of the form k_1+k_2 with k_1>=1, k_2>=1 is 1! So a(2) = 1. For n=3 the largest number not of the form ab+bc+ca (a,b,c >= 1) is believed to be 462 (see A025052).

See A025052 for numbers of form ab+bc+ca (which is the case n=3), A027563 for n=4, A027564 for n=5. See also A027566.

A027566 Number of numbers not of form k_1 k_2 .. k_n (1/k_1 + .. + 1/k_n), k_i >= 1.

Original entry on oeis.org

1, 18, 126, 1652

Offset: 2

R. H. Hardin

Cf. A025052, A027563, A027565, A027565.

Showing 1-10 of 22 results. Next

cp's OEIS Frontend

A000926 Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers).

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PARI

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A093669 Numbers having a unique representation as ab+ac+bc, with 0 < a < b < c.

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Python

A025051 Numbers of the form j*k + k*i + i*j, where i,j,k >= 1.

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Formula

A027563 Numbers not of form abc + abd + acd + bcd for 1<=a<=b<=c<=d.

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A094376 Least number having exactly n representations as ab+ac+bc with 0 < a < b < c.

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Maple

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A094377 Greatest number having exactly n representations as ab+ac+bc with 0 < a < b < c.

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Mathematica

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A094379 Least number having exactly n representations as ab+ac+bc with 1 <= a <= b <= c.

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Haskell

Mathematica

A027564 Numbers not of form abcd + abce + abde + acde + bcde for 1 <= a <= b <= c <= d <= e.

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A025051 Numbers of the form jk + ki + i*j, where i,j,k >= 1.