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 A279864 Irregular triangle read by rows: the n-th row corresponds to the natural numbers not exceeding A002110(n) and divisible by the n-th prime but not by a smaller prime. 4
 2, 3, 5, 25, 7, 49, 77, 91, 119, 133, 161, 203, 11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583, 649, 671, 737, 781, 803, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1331, 1397, 1441, 1507, 1529, 1573, 1639, 1661, 1727, 1793, 1837, 1859 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The n-th row has A005867(n-1) terms. The n-th row starts with the n-th prime. The terms of this sequence appear, in that order, while applying the sieve of Eratosthenes; the n-th rows matches the first A005867(n-1) terms of the n-th row of A083140. Any number n>1 can be uniquely written as n = T(i,j)+k*A002110(i) (with k>=0); in that case: - i = A055396(n), - k = floor( (n-1)/A002110(A055396(n)) ). This sequence corresponds to the numbers n>1 such that n <= A002110(A055396(n)). Let S(i,j) = { T(i,j)+k*A002110(i) with k>=0 }, then: - For any n>0, { S(n,j) } is a partition of the numbers divisible by the n-th prime but not by a smaller prime, - For any n>0, { S(i,j) such that i<=n } is a partition of the numbers divisible by the n-th prime, - { S(i,j) } is a partition of the numbers > 1. LINKS Rémy Sigrist, Table of n, a(n) for n = 1..6300 [First 7 rows] FORMULA T(n,1) = A000040(n) for any n>0. T(n,k) = A083140(n,k) for any n>0 and k<=A005867(n-1). EXAMPLE From M. F. Hasler, May 16 2017: (Start) The triangle starts 2; 3; 5, 25; 7, 49, 77, 91, 119, 133, 161, 203; 11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583, 649, 671, 737, 781, 803, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1331, 1397, 1441, 1507, 1529, 1573, 1639, 1661, 1727, 1793, 1837, 1859, 1903, 1969, 1991, 2057, 2101, 2123, 2167, 2189, 2299; ... (End) MATHEMATICA Table[Function[p, Select[Range[Times @@ p], Function[k, And[Divisible[k, Last@ p], Total@ Boole@ Divisible[k, Most@ p] == 0]]]]@ Prime@ Range@ n, {n, 5}] // Flatten (* Michael De Vlieger, Dec 21 2016 *) a[1] = {2}; a[2] = {3}; t[2] = {1, 5}; a[n_] := a[n] = Prime[n]*t[n - 1]; t[x_] := t[x] = Complement[Flatten[Table[k*Product[Prime[j], {j, x - 1}] + t[x - 1], {k, 0, Prime[x] - 1}]], a[x]]; Flatten[Table[a[n], {n, 6}]] (* L. Edson Jeffery, May 16 2017 *) PROG (PARI) pp=1; for (r=1, 5, forstep(n=prime(r), pp*prime(r), prime(r), if (gcd(n, pp)==1, print1 (n ", "))); pp *= prime(r); print("")) (PARI) A279864_row(r, p=prime(r), P=prod(i=1, r-1, prime(i)))=select(n->gcd(n, P)==1, p*[1..P]) \\ M. F. Hasler, May 16 2017 CROSSREFS Cf. A002110, A005867, A083140, A055396, A000040. Sequence in context: A084730 A117696 A256463 * A324289 A280258 A181730 Adjacent sequences: A279861 A279862 A279863 * A279865 A279866 A279867 KEYWORD nonn,tabf AUTHOR Rémy Sigrist, Dec 21 2016 STATUS approved

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Last modified June 9 23:59 EDT 2023. Contains 363183 sequences. (Running on oeis4.)