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%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
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