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A125959 Infinite array of nine columns, read by rows: A(j,k) = digital root of j*k for j >= 1, 1 <= k <= 9. 0

%I

%S 1,2,3,4,5,6,7,8,9,2,4,6,8,1,3,5,7,9,3,6,9,3,6,9,3,6,9,4,8,3,7,2,6,1,

%T 5,9,5,1,6,2,7,3,8,4,9,6,3,9,6,3,9,6,3,9,7,5,3,1,8,6,2,4,9,8,7,6,5,4,

%U 3,2,1,9,9,9,9,9,9,9,9,9,9,1,2,3,4,5,6

%N Infinite array of nine columns, read by rows: A(j,k) = digital root of j*k for j >= 1, 1 <= k <= 9.

%C a(n) = digital root of (1+floor(n/9))*(1+((n-1) mod 9)).

%C Sequence is periodic, period length is 81. The 9 by 9 array of the first 81 terms is symmetric.

%C To determine the digital root of a product a*b by means of this sequence (or rather by means of the 9 by 9 array) reduce both a and b modulo 9 - if a result is zero replace it by 9 - to obtain c and d, then choose the d-th element of the c-th row, or alternatively the c-th element of the d-th row, i.e. A(c,d) or A(d,c); commutativity of multiplication is reflected in the symmetry of the array.

%e Array begins:

%e 1 2 3 4 5 6 7 8 9

%e 2 4 6 8 1 3 5 7 9

%e 3 6 9 3 6 9 3 6 9

%e 4 8 3 7 2 6 1 5 9

%e 5 1 6 2 7 3 8 4 9

%e 6 3 9 6 3 9 6 3 9

%e 7 5 3 1 8 6 4 2 9

%e 8 7 6 5 4 3 2 1 9

%e 9 9 9 9 9 9 9 9 9

%e Find the digital root of 197*799

%e (a) customarily: digital root of 197*799 = digital root of (digital root of 197)*(digital root of 799) = digital root of 8*7 = digital root of 56 = 2.

%e (b) using the array: 197 mod 9 = 8, 799 mod 9 = 7; A(8,7) = A(7,8) = 2.

%o (PARI) {digitalroot(n) = if(n<1, 0, (n-1)%9+1)} {a(n) = digitalroot((1+floor(n/9))*(1+((n-1)%9)))} {for(n=1, 105, print1(a(n), ","))} /* Klaus Brockhaus, Mar 28 2007 */

%Y Cf. A010888 (digital root of n).

%K nonn,tabf,base,easy

%O 1,2

%A Simon Alexander (simonalexander2005(AT)hotmail.com), Feb 07 2007

%E Edited by _Klaus Brockhaus_, Mar 28 2007

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Last modified July 17 16:38 EDT 2019. Contains 325107 sequences. (Running on oeis4.)