A096135 Triangle read by rows which contains in row n that permutation of the n numbers T(n-1)+1..T(n) which yields a smallest multiple of n after concatenation. T(n) are the triangular numbers. If no such multiple exists, the row contains zeros.

1, 3, 2, 4, 5, 6, 7, 9, 10, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 20, 22, 23, 24, 25, 28, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 50, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset

1,2

Comments

Conjecture: No term is zero.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..190 (rows 1..19)

Example

Triangle begins:
  1;
  3,2;
  4,5,6;
  7,9,10,8;
  11,12,13,14,15;
  16,17,18,19,21,20;
  ...
Terms a(7) to a(10) are 7,9,10,8 respectively, a permutation of 7,8,9 and 10, the next four numbers and 79108 is a multiple of 4.
The 4th row contains a permutation of 7, 8, 9 and 10, the four numbers up to and including T(4)=A000217(4)=10. The concatenations 78910, 78109 etc. are not multiples of 4, which leaves the concatenation 79108 as the only (and therefore minimal) candidate for the ordering of the numbers in the row.
Row 14 is (100, 101, 102, 103, 104, 105, 92, 93, 94, 95, 97, 98, 99, 96). This is the first case where the smallest multiple is not the lexically least permutation whose concatenation is a multiple of n. The lexically least permutation would be (92, 93, 94, 95, 96, 97, 98, 99, 100, 103, 101, 105, 102, 104). - Andrew Howroyd, Aug 09 2025

Prog

(PARI) \\ suitable up to row 19. Too slow on row 20, 40, 50.
row(n)={my(v=vecsort(vector(n, i, digits(binomial(n, 2)+i)))); forperm(n, u, my(m=fromdigits(concat(vector(n, i, v[u[i]])))); if(m%n==0, return( vector(n, i, fromdigits(v[u[i]])) ) ))} \\ Andrew Howroyd, Aug 09 2025

Crossrefs

Cf. A096136.

Sequence in context: A171086 A333324 A333470 * A092829 A081943 A357872

Adjacent sequences: A096132 A096133 A096134 * A096136 A096137 A096138

Keyword

base,nonn,tabl

Author

Amarnath Murthy, Jul 06 2004

Extensions

More terms from R. J. Mathar, Aug 03 2007

a(46) onwards from Andrew Howroyd, Aug 09 2025

Status

approved