A364732 -id:A364732 - cp's OEIS frontend

A364443 a(n) is the number of integers k of the form x^2 + x*y + y^2 (A003136) with n^2 < k < (n+1)^2.

0, 1, 1, 2, 2, 3, 4, 4, 5, 4, 6, 5, 6, 8, 7, 8, 7, 9, 9, 11, 10, 10, 11, 10, 13, 12, 13, 13, 13, 14, 13, 16, 16, 16, 14, 16, 17, 16, 18, 20, 19, 19, 19, 19, 21, 20, 22, 21, 21, 22, 22, 24, 25, 21, 24, 25, 24, 27, 27, 25, 29, 26, 28, 26, 27, 29, 29, 30, 28, 29, 32, 31

Offset: 0

Hugo Pfoertner, Aug 05 2023

nonn

a(n) is the number of circles centered at (0,0) that pass through grid points of the hexagonal lattice that intersect the interior of an interval n < x < n+1 on the x-axis.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
IBM Research, Circles on a triangular grid, Ponder This Challenge December 2023.
Hugo Pfoertner, Table of n, a(n) for n = 0..500000

Cf. A002324, A004016, A077773, A357112, A364729, A364730, A364731, A364732.

is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3);
for (k=0, 75, my (k1=k^2+1, k2=k^2+2*k, m=0); for (j=k1, k2, m+=is_a003136(j)); print1(m,", "))

from sympy import factorint
def A364443(n): return sum(1 for k in range(n**2+1,(n+1)**2) if not any(e&1 for p, e in factorint(k).items() if p % 3 == 2)) # Chai Wah Wu, Aug 07 2023

A364729 Complement of A364443.

Original entry on oeis.org

15, 23, 45, 53, 66, 78, 136, 144, 162, 184, 191, 208, 261, 265, 269, 310, 337, 458, 476, 539, 550, 557, 594, 614, 667, 681, 766, 772, 785, 806, 808, 863, 870, 879, 896, 910, 923, 927, 942, 975, 992, 1012, 1013, 1050, 1053, 1066, 1071, 1154, 1193, 1223, 1254, 1271

Offset: 1

Hugo Pfoertner, Aug 06 2023

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..2340

Cf. A363762, A364445, A364730, A364731, A364732.

A364730(a(n)) = 0.

A364731(a(n)) = A364732(a(n)) = -1.

A364730 a(n) is the number of occurrences of n in A364443.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 2, 2, 2, 2, 3, 2, 1, 5, 2, 0, 5, 1, 1, 4, 2, 4, 3, 0, 3, 3, 2, 3, 2, 4, 2, 2, 3, 2, 2, 4, 4, 1, 1, 3, 2, 4, 4, 2, 3, 0, 5, 3, 1, 2, 5, 1, 5, 0, 4, 3, 2, 2, 5, 2, 3, 1, 4, 3, 1, 4, 0, 6, 3, 3, 2, 2, 3, 1, 3, 1, 9, 2, 0, 2, 5, 3, 4, 1, 3, 2, 4, 3

Offset: 0

Hugo Pfoertner, Aug 06 2023

nonn

Compared to A364443 this sequence is what A363522 is to A077773.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000

Cf. A077773, A363522, A364729, A364731, A364732.

A364731 a(n) is the index of the first occurrence of n in A364443, or -1 if n doesn't occur.

Original entry on oeis.org

0, 1, 3, 5, 6, 8, 10, 14, 13, 17, 20, 19, 25, 24, 29, -1, 31, 36, 38, 40, 39, 44, 46, -1, 51, 52, 61, 57, 62, 60, 67, 71, 70, 72, 76, 77, 79, 92, 94, 86, 90, 91, 95, 99, 105, -1, 108, 104, 118, 116, 114, 124, 122, -1, 123, 130, 135, 134, 141, 138, 140, 160, 148

Offset: 0

Hugo Pfoertner, Aug 06 2023

sign

Compared to A364443, this sequence is what A363763 is to A077773.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000

Cf. A077773, A363763, A364729, A364730, A364732.

cp's OEIS Frontend

A364443 a(n) is the number of integers k of the form x^2 + x*y + y^2 (A003136) with n^2 < k < (n+1)^2.

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A364729 Complement of A364443.

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A364730 a(n) is the number of occurrences of n in A364443.

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A364731 a(n) is the index of the first occurrence of n in A364443, or -1 if n doesn't occur.

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