A381226 -id:A381226 - cp's OEIS frontend

A381227 Irregular triangle read by rows: row n lists the A381226(n) numbers constructed in the definition of A381226, in increasing order.

1, 1, 2, 1, 2, 3, 6, 1, 2, 3, 4, 5, 24, 1, 2, 3, 4, 10, 11, 120, 1, 2, 3, 5, 6, 26, 27, 720, 1, 2, 3, 8, 9, 70, 71, 5040, 1, 2, 3, 4, 14, 15, 200, 201, 40320, 1, 2, 3, 4, 5, 24, 25, 602, 603, 362880, 1, 2, 3, 6, 7, 43, 44, 1904, 1905, 3628800, 1, 2, 3, 8, 9, 79, 80, 6317, 6318, 39916800

Offset: 1

N. J. A. Sloane, Feb 25 2025

			Triangle begins:
   1;
   1, 2;
   1, 2, 3, 6;
   1, 2, 3, 4,  5, 24;
   1, 2, 3, 4, 10, 11, 120;
   1, 2, 3, 5,  6, 26,  27,  720;
   1, 2, 3, 8,  9, 70,  71, 5040;
   1, 2, 3, 4, 14, 15, 200,  201, 40320;
   1, 2, 3, 4,  5, 24,  25,  602,   603,  362880;
   1, 2, 3, 6,  7, 43,  44, 1904,  1905, 3628800;
   ...

Cf. A139004, A381226, A381228, A381229.

A381228 Smallest k such that n appears in row k of the triangle in A381227, or -1 if n never appears in A381227.

Original entry on oeis.org

1, 2, 3, 4, 4, 3, 10, 7, 7, 5, 5, 12, 12, 8, 8, 13, 13, 21, 36, 22, 22, 37, 14, 4, 9, 6, 6, 39, 39, 24, 24, 15, 15, 69, 41, 41, 25, 25, 42, 42, 72, 72, 10, 10, 43, 16, 16, 74, 128, 44, 44, 75, 130, 76, 76, 27, 27, 77, 77, 134, 134, 78, 46, 46, 17, 17, 79, 79, 28

Offset: 1

N. J. A. Sloane, Feb 25 2025

nonn

Conjecture: Every positive integer appears in A381227.

			14 first appears in row 8 of A381227, so a(14) = 8.

Cf. A139004, A381226, A381227, A381229.

More terms from Jinyuan Wang, Feb 25 2025

A381229 a(n) is the number of distinct positive integers that can be obtained by starting with n, and optionally applying the operations square root, floor, and ceiling, in any order.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 7

Offset: 1

N. J. A. Sloane and Jinyuan Wang, Feb 25 2025

nonn

			For n = 15, sqrt(15) = 3.872..., floor and ceiling give 3 and 4. Sqrt(3) = 1.732..., and floor and ceiling give 1 and 2. 4 gives nothing new. In all, we get a(15) = 5 different numbers: 15, 3, 4, 1, 2.

Cf. A381226, A381227, A381228.

f(n) = my(t); if(n<4, [1..n], t=sqrtint(n); if(issquare(n), concat(f(t), n), Set(concat([f(t), f(t+1), [n]]))));
a(n) = #f(n);

cp's OEIS Frontend

A381227 Irregular triangle read by rows: row n lists the A381226(n) numbers constructed in the definition of A381226, in increasing order.

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A381228 Smallest k such that n appears in row k of the triangle in A381227, or -1 if n never appears in A381227.

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A381229 a(n) is the number of distinct positive integers that can be obtained by starting with n, and optionally applying the operations square root, floor, and ceiling, in any order.

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