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Triangle read by rows: row n contains n terms in increasing order, relatively prime to n, whose sum is a multiple of n and such that the row contains the smallest possible subset of consecutive numbers starting with 1.

5

`%I #16 May 24 2020 04:02:29
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`%S 1,1,3,1,4,7,1,3,5,7,1,2,3,6,8,1,5,7,11,13,17,1,2,3,4,5,8,12,1,3,5,7,
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`%T 9,11,13,15,1,2,4,5,7,8,10,13,22,1,3,7,9,11,13,17,19,21,29,1,2,3,4,5,
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`%U 6,7,8,9,12,20
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`%N Triangle read by rows: row n contains n terms in increasing order, relatively prime to n, whose sum is a multiple of n and such that the row contains the smallest possible subset of consecutive numbers starting with 1.
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`%C The "smallest set of n distinct numbers" is not a well-defined term in the definition. Why is row 5 "1,2,3,6,8" but not "1,2,4,6,7"? Why is row 7 "1,2,3,4,5,8,12" but not "1,2,4,5,6,8,9"? - _R. J. Mathar_, Nov 12 2006
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`%e Triangle begins:
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`%e 1;
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`%e 1, 3;
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`%e 1, 4, 7;
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`%e 1, 3, 5, 7;
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`%e 1, 2, 3, 6, 8;
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`%e 1, 5, 7, 11, 13, 17;
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`%e 1, 2, 3, 4, 5, 8, 12;
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`%e ...
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`%o (PARI) row(n) = {my(m=n*(n-1)/2, v); forstep(k=m+n/(2-n%2), oo, n, v=List([]); for(i=2, k-m, if(gcd(n, i)==1, listput(v, i))); if(#v>n-2, forsubset([#v, n-1], w, if(r=1+sum(i=1, n-1, v[w[i]])==k, return(concat(1, vector(n-1, i, v[w[i]]))))))); } \\ _Jinyuan Wang_, May 24 2020
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`%Y Cf. A081522, A081523, A081524.
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`%K nonn,tabl,more
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`%O 1,3
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`%A _Amarnath Murthy_, Mar 27 2003
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`%E New definition proposed by _Omar E. Pol_, Mar 24 2008, in an attempt to answer _R. J. Mathar_'s questions.
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`%E Name corrected and more terms from _Jinyuan Wang_, May 24 2020
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