A007960 Positive numbers k with the property that some permutation of the digits of k is a triangular number.

1, 3, 6, 10, 12, 15, 19, 21, 28, 30, 36, 45, 51, 54, 55, 60, 63, 66, 78, 82, 87, 91, 100, 102, 105, 109, 117, 120, 123, 132, 135, 136, 147, 150, 153, 156, 163, 165, 168, 171, 174, 186, 190, 201, 208, 210, 213, 231, 235, 253, 258, 267, 276, 280, 285, 300, 306, 307

Offset: 1

Author

R. Muller

Keywords

• nonn
• base

Comments

Leading zeros may be omitted from the permutation of the digits of k to get T. But the number of digits of T must be <= the number of digits of k. - N. J. A. Sloane, Dec 14 2007

Working modulo 9, A010878(A000217(j)) is in the set {0, 1, 3, 6} for all j, never in {2, 4, 5, 7, 8}. Since permutation of decimal digits does not change values mod 9, A010878(n) is also one of {0,1,3,6}. - R. J. Mathar, Jan 08 2008

Examples

			Contains k=1, k=10, k=100, etc. derived from T=1.
Contains k=3, k=30, k=300, etc. derived from T=3.
Contains k=15, k=51, k=105, k=150, etc. derived from T=15.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• F. Smarandache, Only Problems, Not Solutions!

Crossrefs

Cf. A000217.

Programs

Maple

q:= n-> ormap(issqr, map(x-> 1+8*parse(cat(x[])),
        combinat[permute](convert(n, base, 10)))):
select(q, [$1..500])[];  # Alois P. Heinz, Aug 22 2021

Mathematica

Select[Range[500], Length[Select[FromDigits/@Permutations[ IntegerDigits[#]], IntegerQ[(Sqrt[1+8#]-1)/2]&]]>0&]  (* Marco Ripà, Nov 07 2022 *)

Extensions

A large number of errors corrected by N. J. A. Sloane, Apr 15 1996

Edited, corrected and extended by R. J. Mathar, Jan 08 2008