A198682 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3*k+2.

6, 12, 18, 30, 36, 51, 54, 69, 75, 84, 90, 105, 108, 123, 129, 141, 147, 153, 162, 177, 183, 195, 201, 207, 219, 225, 240, 246, 252, 267, 270, 285, 291, 303, 309, 315, 324, 339, 345, 357, 363, 369, 381, 387, 402, 411, 417, 423, 435, 441, 456, 459, 474, 480, 486

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.

For each m, the sequence contains exactly one of 9*m, 9*m+3, 9*m+6. - Robert Israel, Mar 04 2016

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• 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, A198681.

Programs

Maple

select(t -> convert(convert(t,base,3),`+`) mod 3 = 2, [seq(3*i,i=1..1000)]); # Robert Israel, Mar 04 2016

Mathematica

Select[Range[3, 498, 3], IntegerQ[(-2 + Plus@@IntegerDigits[#, 3])/3] &] (* Alonso del Arte, Nov 02 2011 *)

PARI

isok(n) = !(n % 3) && !((vecsum(digits(n, 3)) - 2) % 3); \\ Michel Marcus, Mar 02 2016

Extensions

Offset corrected by Michel Marcus, Mar 02 2016