A198681 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3k+1.

3, 9, 24, 27, 42, 48, 60, 66, 72, 81, 96, 102, 114, 120, 126, 138, 144, 159, 168, 174, 180, 192, 198, 213, 216, 231, 237, 243, 258, 264, 276, 282, 288, 300, 306, 321, 330, 336, 342, 354, 360, 375, 378, 393, 399, 408, 414, 429, 432, 447, 453, 465, 471, 477, 492, 498

Offset: 1

Author

John W. Layman, Oct 28 2011

Keywords

• nonn
• base

Comments

It appears that Sum[k^j, 0<=k<=2^n-1, k in A198680] = Sum[k^j, 0<=k<=2^n-1, k in A198681] = Sum[k^j, 0<=k<=2^n-1, k in A180682], for 0<=j<=n-1, which has been verified numerically in a number of cases. This is a generalization of Prouhet's Theorem (see the reference). To illustrate for j=3, we have Sum[k^3, 0<=k<=2^n-1, k in A198680] = {0,0,12636,1108809,94478400,7780827681,633724260624,51425722195929,4168024588857600,...}, Sum[k^3, 0<=k<=2^n-1, k in A198681] = {0,27,14580,1095687,94478400,7780827681,633724260624,51425722195929,4168024588857600,..., Sum[k^3, 0<=k<=2^n-1, k in A198682] = {0,216,7776,1121931,94478400,7780827681,633724260624,51425722195929,4168024588857600,...}, and it is seen that all three sums agree for n>=4=j+1.

Links

• Michel Marcus, Table of n, a(n) for n = 1..1001 (after Harvey P. Dale).
• Chris Bernhardt, Evil twins alternate with odious twins, Math. Mag. 82 (2009), pp. 57-62.
• Eric Weisstein's World of Mathematics, Prouhet-Tarry-Escott Problem

Crossrefs

Cf. A000069, A001969, A157971, A158704, A198680, A198682.

Programs

Mathematica

Select[3Range[200],IntegerQ[(Total[IntegerDigits[#,3]]-1)/3]&] (* Harvey P. Dale, Feb 05 2012 *)

Extensions

Offset corrected by Michel Marcus, Mar 02 2016