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.

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

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