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 A229758 Triangle T(n,k) = A007953(n) * A007953(k) - A007953(n*k) with 1 <= k <= n. 1

%I

%S 0,0,0,0,0,0,0,0,9,9,0,9,9,18,18,0,9,9,18,27,27,0,9,18,18,27,36,36,0,

%T 9,18,27,36,36,45,54,0,9,18,27,36,45,54,63,72,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,9,9,9,9,18,0,0,0

%N Triangle T(n,k) = A007953(n) * A007953(k) - A007953(n*k) with 1 <= k <= n.

%C 9 divides all terms, which shows the "casting out nines" congruence of multiplication.

%H Indranil Ghosh, <a href="/A229758/b229758.txt">Rows 1..120 of triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CastingOutNines.html">Casting Out Nines</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Digit_sum">Digit sum</a>

%e From _Indranil Ghosh_, Feb 14 2017, (Start)

%e Table:

%e 1: 0,

%e 2: 0, 0,

%e 3: 0, 0, 0,

%e 4: 0, 0, 9, 9,

%e 5: 0, 9, 9, 18, 18,

%e 6: 0, 9, 9, 18, 27, 27,

%e 7: 0, 9, 18, 18, 27, 36, 36,

%e 8: 0, 9, 18, 27, 36, 36, 45, 54,

%e 9: 0, 9, 18, 27, 36, 45, 54, 63, 72,

%e 10: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

%e 11: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

%e 12: 0, 0, 0, 0, 9, 9, 9, 9, 18, 0, 0, 0,

%e ...

%e T(12,5) = A007953(12) * A007953(5) - A007953(12 * 5) = 3 * 5 - 6 = 15 - 6 = 9. (End)

%t dSum[n_] := Total[IntegerDigits[n]]; Flatten[Table[dSum[n]*dSum[k] - dSum[n*k], {n, 12}, {k, n}]] (* _T. D. Noe_, Oct 01 2013 *)

%Y Cf. A007953.

%K nonn,base,tabl

%O 1,9

%A _Enrique PĂ©rez Herrero_, Sep 28 2013

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Last modified August 6 19:52 EDT 2020. Contains 336256 sequences. (Running on oeis4.)