A003273 -id:A003273 - cp's OEIS frontend

A248407 Squarefree noncongruent numbers.

1, 2, 3, 10, 11, 17, 19, 26, 33, 35, 42, 43, 51, 57, 58, 59, 66, 67, 73, 74, 82, 83, 89, 91, 97, 105, 106, 107, 113, 114, 115, 122, 123, 129, 130, 131, 139, 146, 155, 163, 170, 177, 178, 179, 185, 186, 187, 193, 195, 201, 202, 203, 209

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2580 (terms below 10^4)
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Intersection of A005117 and A165564.

Cf. A003273, A248394, A248395, A248397-A248406.

A147778 Positive integers of the form uv(u^2 - v^2) where u, v are coprime integers.

Original entry on oeis.org

6, 24, 30, 60, 84, 120, 180, 210, 240, 330, 336, 504, 546, 630, 720, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2016, 2184, 2310, 2340, 2520, 2574, 2730, 3036, 3360, 3570, 3696, 3900, 3960, 4080, 4290, 4620, 4896, 4914, 5016, 5280, 5544, 5610, 5814

Offset: 1

Max Alekseyev, Nov 12 2008

nonn

Terms with even u or v form A024365. Squarefree terms form A147779.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of: A003273, A009112, A073120.

N:= 10^5:
A:= {}:
for v from 1 to floor((N/2)^(1/3)) do
   for u from v+1 do
      if igcd(u,v) = 1 then
        t:= u*v*(u^2-v^2);
        if t > N then break fi;
        A:= A union {t};
      fi
    od
od:
A;
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 06 2015

A147779 Squarefree positive integers of the form uv(u^2-v^2) for some integer u,v.

Original entry on oeis.org

6, 30, 210, 330, 546, 2310, 2730, 3570, 4290, 5610, 6090, 6630, 7854, 8970, 9690, 10374, 10626, 13566, 18354, 19866, 22134, 25806, 26970, 39270, 43890, 51330, 51414, 52026, 54834, 56730, 59334, 66990, 68034, 71610, 72930, 74046, 75174

Offset: 1

Max Alekseyev, Nov 12 2008

nonn

Cf. A003273, A006991, A009112, A073120, A147778.

Squarefree terms of A147778. Squarefree terms of A073120. Squarefree terms of A009112.

Terms of A006991 (primitive congruent numbers) corresponding to right triangles with integer sides.

A364202 Integers m which can be written as m = pq = rs, with 1 <= r < p < q < s <= m and satisfying (p+q) | (s-r).

Original entry on oeis.org

6, 21, 24, 30, 40, 52, 54, 60, 72, 84, 96, 105, 120, 126, 150, 154, 160, 165, 180, 186, 189, 204, 208, 210, 216, 240, 270, 273, 288, 294, 300, 301, 312, 322, 330, 336, 342, 357, 360, 378, 384, 414, 420, 456, 468, 480, 486, 504, 525, 540, 546, 550, 594, 600

Offset: 1

Jose Aranda, Jul 13 2023

nonn

Terms may have multiple solutions p,q,r,s, and each has a least quotient k = (s-r) / (p+q).
Those with k=1 are the congruent numbers (A003273) and others are a more general case.
They all share a simple inter-square characterization. The 4 squares are A = (q-p)^2, B = (p+q)^2, C = ((p+q)*k)^2 and D = (r+s)^2. We have B = A + 4m, C = B*(k^2) and D = C + 4m, where 4m is added exclusively to avoid the use of fractions.

			21 is a term since 21 = 3*7 = 1*21 which has 3+7 = 10 divides 21-1 = 20 (k=2).
So there are 4 squares, in this case, 16, 100, 400 and 484, which are related by this number. In effect, 4*21=+84 jumps from the first to the second, which, multiplied by k^2, gives the third, where +84 gives the fourth.

Jose Aranda, Table of n, a(n) for n = 1..9999

Cf. A003273 (congruent numbers).

isok(k) = my(d=divisors(k)); if (#d >= 4, for (i=1, #d-1, my(r = d[i], s = k/r); if (rMichel Marcus, Jul 17 2023

More terms from Alois P. Heinz, Jul 13 2023

A165816 Prime congruent numbers (A165815) that are not equal to 5 or 7 (mod 8).

Original entry on oeis.org

41, 137, 257, 313, 353, 457, 761, 1201, 1217, 1249, 1321, 2113, 2273, 2777, 2833, 2953, 3001, 3433, 3593, 3761, 3881, 4441, 4481, 4649, 4793, 4889, 5273, 5449, 5569, 5657, 5849, 6073, 6529, 7001, 7321, 7417, 7561, 7793, 8521, 8609, 9049, 9257, 9281

Offset: 1

T. D. Noe, Sep 28 2009

nonn

Heegner proved that every prime p with p = 5 or 7 (mod 8) is a congruent number. See A003628 for those primes. All primes in this sequence equal 1 (mod 8).

Monsky proved that no prime of the form 8k+3 is a congruent number. - Jonathan Sondow, Nov 15 2017

T. D. Noe, Primes less than 10^7
Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Zeitschrift 56 (1952), 227-253.
P. Monsky, Mock Heegner points and congruent numbers, Math. Zeit., 204 (1990), 45-67.
Kent E. Morrison, Congruent Numbers

Cf. A003273 (congruent numbers), A165815 (prime congruent numbers).

A257604 Congruentful numbers. Numbers that are not congruentfree. Complement of A253278.

Original entry on oeis.org

5, 6, 7, 10, 12, 13, 14, 15, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 100

Offset: 1

Juri-Stepan Gerasimov, May 06 2015

These are the positive integers divisible by a congruent number given in A003273. - Wolfdieter Lang, May 09 2015

			10 is in this sequence because the positive integer 10 is divisible by congruent number 5.

Cf. A003273, A253278.

A260960 Least positive integer k < prime(n) such that there are 0 < i < j with i^2 + j^2 = k^2 for which (i*j)/2 is a primitive root modulo prime(n), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 5, 5, 5, 17, 13, 17, 10, 10, 5, 13, 13, 25, 5, 5, 39, 25, 17, 5, 5, 5, 17, 29, 5, 5, 5, 5, 5, 5, 5, 34, 17, 5, 5, 26, 13, 13, 5, 10, 29, 13, 13, 5, 34, 5, 5, 5, 5, 25, 25, 5, 5, 13, 17, 5, 5, 10, 29, 13, 13, 61, 17, 13, 17, 17, 5, 13

Offset: 1

Zhi-Wei Sun, Aug 06 2015

nonn

Conjecture: a(n) > 0 for any n > 4. In other words, for any prime p > 7, there exists a right triangle whose three sides are among 1,...,p-1 and whose area is a primitive root modulo p.
We have verified this for primes p < 10^5.
We also conjecture that for any prime p > 31, there exists a right triangle whose three sides are among 1,...,p-1, and whose perimeter and area are quadratic residues modulo p.

			a(7) = 5 since 3^2 + 4^2 = 5^2, and (3*4)/2 = 6 is a primitive root modulo prime(7) = 17.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A003273, A260911, A260946, A260947.

SQ[n_]:=IntegerQ[Sqrt[n]]
Dv[n_]:=Divisors[Prime[n]-1]
Do[Do[Do[If[SQ[k^2-j^2]==False, Goto[cc]];Do[If[Mod[(j*Sqrt[k^2-j^2]/2)^(Part[Dv[n],t]),Prime[n]]==1,Goto[cc]];Continue,{t,1,Length[Dv[n]]-1}];
Print[n," ",k];Goto[aa];Label[cc];Continue,{j,1,k-1}];Label[dd];Continue,{k,1,Prime[n]-1}];Print[n," ",0];Label[aa];Continue,{n,1,70}]

A248398 Noncongruent squarefree numbers n with A248394(n)/d(n) = -1, where d(n) = A000005(n).

Original entry on oeis.org

11, 19, 35, 67, 91, 105, 115, 123, 129, 179, 195, 201, 227, 235, 249, 273, 347, 393, 403, 419, 427, 435, 473, 483, 563, 611, 635, 683, 691, 705, 715, 739, 753, 779, 787, 795, 817, 843, 851, 993, 1051, 1115, 1121, 1123, 1177, 1209, 1265, 1347, 1401, 1435, 1441

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A165564, A248394, A248395, A248397-A248407.

More terms from Amiram Eldar, Oct 13 2019

A248399 Noncongruent squarefree numbers n with A248394(n)/d(n) = 2, where d(n) = A000005(n).

Original entry on oeis.org

73, 155, 185, 203, 241, 281, 329, 355, 545, 553, 579, 601, 627, 641, 697, 755, 763, 785, 865, 937, 1097, 1139, 1193, 1227, 1243, 1289, 1299, 1353, 1371, 1457, 1465, 1537, 1721, 1753, 1763, 1841, 1865, 1913, 1937, 1961, 2017, 2041, 2105, 2177, 2281, 2307, 2353

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A248394, A248395, A248397-A248406.

More terms from Amiram Eldar, Oct 13 2019

A248400 Noncongruent squarefree numbers n with A248394(n)/d(n) = -2, where d(n) = A000005(n).

Original entry on oeis.org

17, 89, 97, 193, 217, 233, 259, 305, 377, 401, 449, 481, 497, 617, 667, 713, 745, 769, 897, 929, 955, 977, 979, 1009, 1011, 1027, 1033, 1049, 1065, 1337, 1345, 1355, 1385, 1409, 1417, 1489, 1507, 1555, 1739, 1769, 1771, 1801, 1803, 1817, 1921, 1945, 2001, 2019

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A248394, A248395, A248397-A248406.

More terms from Amiram Eldar, Oct 13 2019

cp's OEIS Frontend

A248407 Squarefree noncongruent numbers.

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A147778 Positive integers of the form u*v*(u^2 - v^2) where u, v are coprime integers.

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Maple

A147779 Squarefree positive integers of the form u*v*(u^2-v^2) for some integer u,v.

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A364202 Integers m which can be written as m = p*q = r*s, with 1 <= r < p < q < s <= m and satisfying (p+q) | (s-r).

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A165816 Prime congruent numbers (A165815) that are not equal to 5 or 7 (mod 8).

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A257604 Congruentful numbers. Numbers that are not congruentfree. Complement of A253278.

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A260960 Least positive integer k < prime(n) such that there are 0 < i < j with i^2 + j^2 = k^2 for which (i*j)/2 is a primitive root modulo prime(n), or 0 if no such k exists.

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Mathematica

A248398 Noncongruent squarefree numbers n with A248394(n)/d(n) = -1, where d(n) = A000005(n).

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A248399 Noncongruent squarefree numbers n with A248394(n)/d(n) = 2, where d(n) = A000005(n).

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A248400 Noncongruent squarefree numbers n with A248394(n)/d(n) = -2, where d(n) = A000005(n).

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Extensions

A147778 Positive integers of the form uv(u^2 - v^2) where u, v are coprime integers.

A147779 Squarefree positive integers of the form uv(u^2-v^2) for some integer u,v.

A364202 Integers m which can be written as m = pq = rs, with 1 <= r < p < q < s <= m and satisfying (p+q) | (s-r).