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 A299714 Irregular triangle read by rows: row n contains numbers k such that 1<=k<=2*n+1 and gcd(n-k, 2*n+1) != 1. 1
 1, 2, 3, 1, 4, 7, 5, 6, 1, 2, 4, 7, 10, 12, 13, 8, 9, 1, 3, 4, 7, 10, 13, 16, 17, 19, 11, 2, 7, 12, 17, 22, 1, 4, 7, 10, 13, 16, 19, 22, 25, 14, 15, 1, 4, 5, 7, 10, 13, 16, 19, 22, 25, 27, 28, 31, 2, 3, 7, 10, 12, 17, 22, 24, 27, 31, 32, 18, 1, 4, 6, 7, 10, 13, 16, 19, 22, 25, 28, 31, 32, 34, 37, 20, 21, 1, 2, 4, 7, 10, 12, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row n contains 2*n+1 - phi(2*n+1) = A053193(n) terms. Row n has just one term (namely n) if 2*n+1 is prime. If we extend each row [r,s, ..., z] to infinite length by appending [j*N*r,j*N*s, ..., j*N*z] for j>=1 and N=2*n+1 we obtain a relation on pairs of positive numbers (u,v). Write R(u,v) if v appears in row u (the relation is 1 on the pair (u,v)). Then R(u,u), R(u,v) ==> R(v, u), R(u,v) ==> R(u, v mod (2*u+1) ), and R(u,v) and v > u ==> R(u-(v-u), v-(v-u)) = R(2*u - v, u). LINKS FORMULA T(n,k) = A300288(n,k) - n. EXAMPLE Triangle starts: [01]: [1] [02]: [2] [03]: [3] [04]: [1, 4, 7] [05]: [5] [06]: [6] [07]: [1, 2, 4, 7, 10, 12, 13] [08]: [8] [09]: [9] [10]: [1, 3, 4, 7, 10, 13, 16, 17, 19] [11]: [11] [12]: [2, 7, 12, 17, 22] [13]: [1, 4, 7, 10, 13, 16, 19, 22, 25] [14]: [14] [15]: [15] [16]: [1, 4, 5, 7, 10, 13, 16, 19, 22, 25, 27, 28, 31] [17]: [2, 3, 7, 10, 12, 17, 22, 24, 27, 31, 32] [18]: [18] ... MAPLE T:= n-> select(k-> igcd(n-k, 2*n+1)<>1, [\$1..2*n+1])[]: seq(T(n), n=1..25); # Alois P. Heinz, Mar 09 2018 PROG (PARI) is(n, k)= ( gcd(n-k, 2*n+1)!=1 ); for (n=1, 33, for (k=1, 2*n+1, if ( is(n, k), print1(k, ", ") ); ); ); CROSSREFS Cf. A300288. Sequence in context: A356769 A111776 A189187 * A171083 A258579 A263757 Adjacent sequences: A299711 A299712 A299713 * A299715 A299716 A299717 KEYWORD nonn,tabf AUTHOR Joerg Arndt, Feb 17 2018 STATUS approved

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Last modified March 24 08:50 EDT 2023. Contains 361463 sequences. (Running on oeis4.)