A337111 -id:A337111 - cp's OEIS frontend

A337110 Number of length three 1..n vectors that contain their geometric mean.

1, 2, 3, 10, 11, 12, 13, 20, 33, 34, 35, 42, 43, 44, 45, 64, 65, 78, 79, 86, 87, 88, 89, 96, 121, 122, 135, 142, 143, 144, 145, 164, 165, 166, 167, 198, 199, 200, 201, 208, 209, 210, 211, 218, 231, 232, 233, 252, 289, 314, 315, 322, 323, 336, 337, 344, 345, 346

Offset: 1

Hywel Normington, Aug 16 2020

From David A. Corneth, Aug 26 2020: (Start)
If x^2 == 0 (mod n) has only 1 solution then a(n) = a(n-1) + 1. Proof:
Let (a, b, n) be such a tuple. Let without loss of generality b be the geometric mean of the tuple. Then a*b*n = b^3 and as b is not 0 we have b^2 = a*n. So then b^2 == 0 (mod n). If b^2 == 0 (mod n) has only 1 solution then b = n. This gives the tuple (n, n, n) which has 1 permutation. So giving a(n) = a(n-1) + 1. (End)

			For n = 2, the a(2) = 2 solutions are: (1,1,1) and (2,2,2).
For n = 4, the a(4) = 10 solutions are: (1,1,1),(2,2,2),(3,3,3),(4,4,4) and the 6 permutations of (1,2,4).

Hywel Normington, Python code, 2020.
Hywel Normington, Julia code, 2023.

Cf. A120486, A248434, A337111, A013929, A057918, A000188.

first(n) = {my(s = 0, res = vector(n)); for(i = 1, n, s+=b(i); res[i] = s ); res }
b(n) = { my(s = factorback(factor(n)[, 1]), res = 1); for(i = 1, n \ s - 1, c = (s*i)^2/n; if(denominator(c) == 1 && c <= n, res+=6; ) ); res } \\ David A. Corneth, Aug 26 2020

a(n) = a(n-1) + 1 + 6*A057918(n).

A337559 Number of length three 1..n vectors that contain their harmonic mean.

Original entry on oeis.org

1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 36, 37, 38, 51, 52, 53, 66, 67, 80, 81, 82, 83, 96, 97, 98, 99, 112, 113, 138, 139, 140, 141, 142, 155, 168, 169, 170, 171, 184, 185, 210, 211, 212, 237, 238, 239, 252, 253, 254, 255, 256, 257, 270, 271, 284, 285, 286, 287, 324, 325, 326, 339

Offset: 1

Hywel Normington, Aug 31 2020

nonn

			For n = 1, the only solution is (1,1,1).
For n = 6, the a(6) = 18 solutions are (k,k,k) for k=1,..,6, the 6 permutations of (2,3,6) and the 6 permutations of (3,4,6).
For n = 40, the a(40)-a(39) = 13 new solutions are (40,40,40), the 6 permutations of (10,16,40) and the 6 permutations of (24,30,40).

Hywel Normington, Python code, 2020.

Cf. A248434, A337110, A337111, A337560.

Empirical: If A174903(n) = 0, a(n) = a(n-1) + 1.

a(n)-a(n-1) = 1 (mod 6).

A337560 Number of length four 1..n vectors that contain their harmonic mean.

Original entry on oeis.org

1, 2, 3, 28, 29, 78, 79, 104, 105, 154, 155, 252, 253, 254, 351, 376, 377, 426, 427, 620, 717, 718, 719, 912, 913, 914, 915, 1084, 1085, 1374, 1375, 1400, 1425, 1426, 1619, 1812, 1813, 1814, 1839, 2128, 2129, 2442, 2443, 2564, 2829, 2830, 2831, 3192, 3193

Offset: 1

Hywel Normington, Aug 31 2020

nonn

			For n = 1, the a(1) = 1 solution is (1,1,1,1).
For n = 4, the a(4) = 28 solutions are (k,k,k,k) for k=1,2,3,4, the 12 permutations of (1,2,4,4) and the 12 permutations of (2,3,4,4).
For n = 40, the a(40) - a(39) = 289 new solutions are:
  the 24 permutations of (4, 8, 10, 40),
  the 12 permutations of (4, 10, 40, 40),
  the 12 permutations of (5, 12, 40, 40),
  the 24 permutations of (8, 10, 12, 40),
  the 24 permutations of (8, 15, 20, 40),
  the 12 permutations of (10, 16, 16, 40),
  the 24 permutations of (10, 18, 24, 40),
  the 12 permutations of (10, 20, 40, 40),
  the 24 permutations of (12, 20, 24, 40),
  the 24 permutations of (14, 24, 35, 40),
  the 24 permutations of (15, 24, 30, 40),
  the 12 permutations of (16, 16, 20, 40),
  the 12 permutations of (20, 20, 24, 40),
  the 12 permutations of (20, 30, 40, 40),
  the 12 permutations of (24, 30, 30, 40),
  the 12 permutations of (28, 35, 40, 40),
  the 12 permutations of (30, 36, 40, 40),
  and (40, 40, 40, 40).

Hywel Normington, Python code, 2020.

Cf. A248434, A337110, A337111, A337559.

a(n) - a(n-1) = 1 (mod 12).

a(43)-a(49) from Alois P. Heinz, Feb 04 2023

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A337110 Number of length three 1..n vectors that contain their geometric mean.

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A337559 Number of length three 1..n vectors that contain their harmonic mean.

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A337560 Number of length four 1..n vectors that contain their harmonic mean.

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