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

Author

Rémy Sigrist, Dec 21 2016

Keywords

• nonn
• tabf

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.

Examples

			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)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..6300 [First 7 rows]

Crossrefs

Cf. A002110, A005867, A083140, A055396, A000040.

Programs

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 *)

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

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).