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),
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]
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
KEYWORD
nonn,tabf
AUTHOR
Rémy Sigrist, Dec 21 2016
STATUS
approved