A238280 - OEIS

%I #28 Apr 02 2017 21:28:50

%S 0,1,0,2,2,1,1,3,0,3,4,4,1,4,2,5,2,2,3,6,0,4,6,7,5,4,3,7,5,6,3,7,2,6,

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

%U 3,7,4,5,3,7,6,5,3,4,6,10,0,5,8,10,9,9,3,8,7,9,7,12,2,7,10,7,6,11,5,10,4,6,9,14,4,4,7,9,8,13,2,7,6,8,11

%N Irregular triangle read by rows, T(n,k) = Sum_{i = 1..n} k mod i, k = 1..m where m = lcm(1..n).

%C Row n contains A003418(n) terms.

%C The penultimate term (the one before zero) of row n = A000217(n-1).

%H Antti Karttunen, <a href="/A238280/b238280.txt">Rows 1..10 of the table, flattened</a>

%H Kival Ngaokrajang, <a href="/A238280/a238280.pdf">Illustration for n = 1..10</a>

%e Row n of this irregular triangle is obtained by taking the first A003418(n) = lcm(1..n) terms (up to and including the first zero) of the following array (which starts at row n=1 and column k=1 and is periodic in each row):

%e   0; 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0

%e   1  0; 1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0

%e   2  2  1  1  3  0; 2  2  1  1  3  0  2  2  1  1  3  0  2  2 # A110269

%e   3  4  4  1  4  2  5  2  2  3  6  0; 3  4  4  1  4  2  5  2

%e   4  6  7  5  4  3  7  5  6  3  7  2  6  8  4  2  6  5  9  2

%e   5  8 10  9  9  3  8  7  9  7 12  2  7 10  7  6 11  5 10  4

%e   6 10 13 13 14  9  8  8 11 10 16  7 13 10  8  8 14  9 15 10

%e   7 12 16 17 19 15 15  8 12 12 19 11 18 16 15  8 15 11 18 14

%e   8 14 19 21 24 21 22 16 12 13 21 14 22 21 21 15 23 11 19 16

%e   9 16 22 25 29 27 29 24 21 13 22 16 25 25 26 21 30 19 28 16

%o (Small Basic)

%o For n = 1 to 20

%o   k = 1

%o   loop:

%o   rs = 0

%o   For i = 1 To n

%o     rs = rs + math.Remainder(k,i)

%o   EndFor

%o   TextWindow.Write(rs+", ")

%o   If rs > 0 then

%o     k = k + 1

%o     Goto loop

%o   EndIf

%o EndFor

%o (Scheme)

%o (define (A238280 n) (A238280tabf (A236857 n) (A236858 n)))

%o (define (A238280tabf n k) (add (lambda (i) (modulo k i)) 1 n))

%o ;; Implements sum_{i=lowlim..uplim} intfun(i):

%o (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

%o ;; _Antti Karttunen_, Feb 27 2014

%Y Cf. A000217, A003418, A173185, A236856, A236857, A236858.

%K nonn,tabf

%O 1,4

%A _Kival Ngaokrajang_, Feb 22 2014