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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.

%I A279864 #26 Mar 02 2018 02:11:18
%S A279864 2,3,5,25,7,49,77,91,119,133,161,203,11,121,143,187,209,253,319,341,
%T A279864 407,451,473,517,583,649,671,737,781,803,869,913,979,1067,1111,1133,
%U A279864 1177,1199,1243,1331,1397,1441,1507,1529,1573,1639,1661,1727,1793,1837,1859
%N 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.
%C A279864 The n-th row has A005867(n-1) terms.
%C A279864 The n-th row starts with the n-th prime.
%C A279864 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.
%C A279864 Any number n>1 can be uniquely written as n = T(i,j)+k*A002110(i) (with k>=0); in that case:
%C A279864 - i = A055396(n),
%C A279864 - k = floor( (n-1)/A002110(A055396(n)) ).
%C A279864 This sequence corresponds to the numbers n>1 such that n <= A002110(A055396(n)).
%C A279864 Let S(i,j) = { T(i,j)+k*A002110(i) with k>=0 }, then:
%C A279864 - 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,
%C A279864 - For any n>0, { S(i,j) such that i<=n } is a partition of the numbers divisible by the n-th prime,
%C A279864 - { S(i,j) } is a partition of the numbers > 1.
%H A279864 Rémy Sigrist, <a href="/A279864/b279864.txt">Table of n, a(n) for n = 1..6300</a> [First 7 rows]
%F A279864 T(n,1) = A000040(n) for any n>0.
%F A279864 T(n,k) = A083140(n,k) for any n>0 and k<=A005867(n-1).
%e A279864 From _M. F. Hasler_, May 16 2017: (Start)
%e A279864 The triangle starts
%e A279864 2;
%e A279864 3;
%e A279864 5, 25;
%e A279864 7, 49, 77, 91, 119, 133, 161, 203;
%e A279864 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;
%e A279864 ... (End)
%t A279864 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 *)
%t A279864 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 *)
%o A279864 (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(""))
%o A279864 (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
%Y A279864 Cf. A002110, A005867, A083140, A055396, A000040.
%K A279864 nonn,tabf
%O A279864 1,1
%A A279864 _Rémy Sigrist_, Dec 21 2016