A267137 -id:A267137 - cp's OEIS frontend

A267434 Numbers of the form x^2 + x + x*y + y + y^2 (A267137) that are not of the form a^2 + b^2 + c^2 where x, y, a, b and c are integers.

60, 92, 112, 124, 156, 220, 240, 252, 284, 316, 380, 412, 444, 476, 496, 508, 540, 604, 624, 732, 752, 764, 796, 880, 892, 956, 960, 1008, 1020, 1084, 1136, 1180, 1212, 1244, 1264, 1276, 1308, 1340, 1392, 1436, 1472, 1500, 1520, 1532, 1564, 1596, 1692, 1724, 1776, 1792, 1820, 1852, 1884, 1916, 1980, 1984

Offset: 1

Altug Alkan, Jan 15 2016

nonn

Intersection of A004215 and A267137.
Inspiration was the equation x^2 + x + x*y + y + y^2 = a^2 + b^2 + c^2 where x, y, a, b and c are integers.
Complement of this sequence is 0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 14, 16, 17, 20, 21, 22, 24, 25, 26, 30, 32, 33, 34, 36, 37, 40, 41, 42, 44, 46, 49, 50, 52, 54, 56, 57, 58, 64, 65, 66, 69, 70, 72, 74, 76, 80, 81, 82, 85, 86, 89, 90, 94, 96, ...

			60 is a term because 60 = 6^2 + 6 + 6*2 + 2 + 2^2 and there is no integer values of a, b and c for the equation 60 = a^2 + b^2 + c^2.
50 is not a term because 50 = 6^2 + 6 + 6*1 + 1 + 1^2 = 3^2 + 4^2 + 5^2.

Cf. A004215, A267137.

Select[Range@ 2000, And[Resolve[Exists[{x, y}, Reduce[# == x^2 + x + x y + y + y^2, {x, y}, Integers]]], !Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]]] &] (* Michael De Vlieger, Jan 15 2016 *)

isA003136(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
for(n=0, 2000, if(isA003136(3*n+1) && isA004215(n), print1(n, ", ")));

A322430 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^8 is zero.

Original entry on oeis.org

3, 7, 11, 13, 15, 18, 19, 23, 27, 28, 29, 31, 35, 38, 39, 43, 45, 47, 48, 51, 53, 55, 59, 61, 62, 63, 67, 68, 71, 73, 75, 77, 78, 79, 83, 84, 87, 88, 91, 93, 95, 98, 99, 103, 106, 107, 109, 111, 113, 115, 117, 118, 119, 123, 125, 127, 128, 130, 131, 135, 138, 139, 141

Offset: 1

Seiichi Manyama, Dec 07 2018

nonn

Indices of zero entries in A000731.

Complement of A267137. - Kemoneilwe Thabo Moseki, Dec 12 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), this sequence (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

my(x='x+O('x^160)); Vec(select(x->(x==0), Vec(eta(x)^8 - 1), 1)) \\ Michel Marcus, Dec 08 2018

A363675 Numbers k such that the least common multiple of the degrees of the irreducible characters of S_k equals |S_k| = k!.

Original entry on oeis.org

0, 1, 6, 10, 21, 36, 66, 105, 120, 136, 190, 210, 276, 325, 465, 496, 561, 630, 666, 741, 780, 990, 1081, 1176, 1225, 1540, 1596, 1830, 2080, 2145, 2346, 2556, 2926, 3081, 3160, 3240, 3486, 3570, 3916, 4005, 4186, 4560, 4656, 4950, 5050, 5356, 5460, 5886, 6105

Offset: 1

Diego Martin Duro, Jun 14 2023

nonn

Intersection of the sequences of numbers k such that there exists a 2-core partition of k (A267137) and a 3-core partition of k (A000217).

Alois P. Heinz, Table of n, a(n) for n = 1..100
Diego Martin Duro, Arithmetic Properties of Character Degrees and the Generalised Knutson Index, arXiv:2306.12408 [math.RT], 2023.

Cf. A000142, A000217, A010054, A033687, A175595, A260682, A267137, A363676, A363701.

From Alois P. Heinz, Jun 16 2023: (Start)
{ k : A175595(k,2) > 0 and A175595(k,3) > 0 }.
{ k : A010054(k) > 0 and A033687(k) > 0 }. (End)

More terms from Alois P. Heinz, Jun 16 2023

A363676 Numbers k such that the least common multiple of the degrees of the irreducible characters of A_k equals |A_k| = k!/2.

Original entry on oeis.org

0, 1, 2, 5, 6, 8, 10, 12, 17, 21, 30, 36, 57, 66, 80, 105, 120, 122, 136, 190, 192, 210, 212, 233, 276, 302, 325, 380, 408, 465, 496, 530, 561, 597, 630, 632, 666, 705, 741, 780, 782, 822, 905, 990, 992, 1081, 1130, 1176, 1225, 1433, 1540, 1542, 1596, 1772

Offset: 1

Diego Martin Duro, Jun 14 2023

nonn

Intersection of the sequences of numbers k such that there exists a 2-core partition of k or k-2 and a 3-core partition of k. This sequence contains A363675.

			The degrees of the irreducible characters of A_5 are 1,3,3,4,5 so their least common multiple is 5!/2 = 60, so 5 is a term of the sequence.

Diego Martin Duro, Arithmetic Properties of Character Degrees and the Generalised Knutson Index, arXiv:2306.12408 [math.RT], 2023.

Cf. A000217, A010054, A175595, A260682, A267137, A363675, A363701.

More terms from Alois P. Heinz, Jun 16 2023

A270122 Indices of prime numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.

Original entry on oeis.org

1, 3, 7, 12, 13, 24, 25, 26, 33, 35, 45, 50, 51, 53, 55, 59, 60, 77, 79, 80, 82, 84, 87, 88, 104, 106, 113, 116, 121, 123, 127, 135, 136, 140, 148, 152, 159, 165, 169, 174, 176, 178, 184, 186, 189, 204, 209, 211, 212, 216, 218, 221, 223, 226, 227, 237

Offset: 1

Altug Alkan, Mar 11 2016

nonn

How is the distribution of a(n), a(n+1) in this sequence where a(n+1) = a(n) + 1?

			1 is a term because prime(1) = 2 = (-2)^2 + (-2) + (-2)*1 + 1 + 1^2.
3 is a term because prime(3) = 5 = 1^2 + 1 + 1*1 + 1 + 1^2.
7 is a term because prime(7) = 17 = (-5)^2 + (-5) + (-5)*3 + 3 + 3^2.
12 is a term because prime(12) = 37 = (-7)^2 + (-7) + (-7)*5 + 5 + 5^2.
13 is a term because prime(13) = 41 = 7^2 + 7 + 7*(-3) + (-3) + (-3)^2.

Cf. A000040, A003136, A267137.

isA003136(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=1, 250, if(isA003136(3*prime(n)+1), print1(n, ", ")));

cp's OEIS Frontend

A267434 Numbers of the form x^2 + x + x*y + y + y^2 (A267137) that are not of the form a^2 + b^2 + c^2 where x, y, a, b and c are integers.

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A322430 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^8 is zero.

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A363675 Numbers k such that the least common multiple of the degrees of the irreducible characters of S_k equals |S_k| = k!.

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A363676 Numbers k such that the least common multiple of the degrees of the irreducible characters of A_k equals |A_k| = k!/2.

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A270122 Indices of prime numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.

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