A118711 Integers k such that the k-th triangular number t_k has all its base-12 digits contained in {1,5,7,11}.

1, 13, 61, 82, 898, 2962, 2989, 9133, 20077, 20653, 28669, 29266, 35581, 35842, 37501, 99133, 236674, 286717, 424621, 424957, 821698, 941650, 1704301, 1722370, 2978413, 3328258, 4494466, 10022317, 40392829, 49870141, 50668882, 53933053

Offset: 1

Author

Walter Kehowski, May 24 2006

Keywords

• nonn
• base

Comments

In base 12 all primes greater than 3 end in the digits 1, 5, 7, E, where X is 10 and E is 11. They are the digits that satisfy GCD(d,12)=1.

The sequence in base 12 is: 1, 11, 51, 6X, 62X, 186X, 1891, 5351, E751, EE51, 14711, 14E2X, 18711, 188XX, 19851, 49451, E4E6X, 119E11, 185891, 185E11, 33762X, 394E2X, 6X2351, 6E08XX, EE7751, 11460XX, 1608E6X, 3433E51, 1163E591, 14850051, 14E7632X, 1608E311, 18331451, 1870E191, 1974E311, ..., . Note that all elements end in 1 or X. The corresponding triangular numbers after the first end in the digits 17 or 77, but not respectively.

Examples

			82 = 6X_12 is a term since the triangular number t=82*(82+1)/2 = 3403 = 1E77_12.
		

Crossrefs

Cf. A000217, A118706, A119033, A119034.

Programs

Maple

L:=[]: pd:={1,5,7,11}: for w to 1 do for n from 1 to 10^6 do t:=n*(n+1)/2; lod:=convert(t,base,12); sod:=convert(lod,set); if sod subset pd then L:=[op(L), [n,t]] fi; od od; L;

Mathematica

fQ[n_] := Union@ Join[{1, 5, 7, 11}, IntegerDigits[n(n + 1)/2, 12]] == {1, 5, 7, 11}; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]], {n, 10^8}] (* Robert G. Wilson v *)

Formula

k is a term if the k-th triangular number t_k = k*(k+1)/2 has its base-12 digits contained in {1,5,7,11}.

A000217(a(n)) = A118706(n), or equivalently, a(n) = (sqrt(8*A118706(n)+1)-1)/2. - Amiram Eldar, Aug 02 2024

Extensions

Edited and extended (a(23)-a(32)) by Robert G. Wilson v, Jun 20 2006