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A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

%I A083140 #156 Dec 18 2019 08:57:57
%S A083140 2,3,4,5,9,6,7,25,15,8,11,49,35,21,10,13,121,77,55,27,12,17,169,143,
%T A083140 91,65,33,14,19,289,221,187,119,85,39,16,23,361,323,247,209,133,95,45,
%U A083140 18,29,529,437,391,299,253,161,115,51,20,31,841,667,551,493,377,319,203,125,57,22
%N A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).
%C A083140 A permutation of natural numbers >= 2.
%C A083140 The proportion of integers in the n-th row of the array is given by A005867(n-1)/A002110(n) = A038110(n)/A038111(n). - _Peter Kagey_, Jun 03 2019, based on comments by _Jamie Morken_ and discussion with _Tom Hanlon_.
%C A083140 The proportion of the integers after the n-th row of the array is given by A005867(n)/A002110(n). - _Tom Hanlon_, Jun 08 2019
%H A083140 Ivan Neretin, <a href="/A083140/b083140.txt">Table of n, a(n) for n = 2..5051</a>
%H A083140 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H A083140 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%e A083140 Array begins:
%e A083140    2   4   6   8  10  12  14  16  18  20  22  24 .... (A005843 \ {0})
%e A083140    3   9  15  21  27  33  39  45  51  57  63  69 .... (A016945)
%e A083140    5  25  35  55  65  85  95 115 125 145 155 175 .... (A084967)
%e A083140    7  49  77  91 119 133 161 203 217 259 287 301 .... (A084968)
%e A083140   11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)
%e A083140   13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)
%t A083140 a = Join[ {Table[2n, {n, 1, 12}]}, Table[ Take[ Prime[n]*Select[ Range[100], GCD[ Prime[n] #, Product[ Prime[i], {i, 1, n - 1}]] == 1 &], 12], {n, 2, 12}]]; Flatten[ Table[ a[[i, n - i]], {n, 2, 12}, {i, n - 1, 1, -1}]]
%t A083140 (* second program: *)
%t A083140 rows = 12; Clear[T]; Do[For[m = p = Prime[n]; k = 1, k <= rows, m += p, If[ FactorInteger[m][[1, 1]] == p, T[n, k++] = m]], {n, rows}]; Table[T[n - k + 1, k], {n, rows}, {k, n}] // Flatten (* _Jean-François Alcover_, Mar 08 2016 *)
%Y A083140 Cf. A083141 (main diagonal), A083221 (transpose), A004280, A038179, A084967, A084968, A084969, A084970, A084971.
%Y A083140 Arrays of integers grouped into rows by various criteria:
%Y A083140 by greatest prime factor: A125624,
%Y A083140 by lowest prime factor: this sequence (upward antidiagonals), A083221 (downward antidiagonals),
%Y A083140 by number of distinct prime factors: A125666,
%Y A083140 by number of prime factors counted with multiplicity: A078840,
%Y A083140 by prime signature: A095904,
%Y A083140 by ordered prime signature: A096153,
%Y A083140 by number of divisors: A119586,
%Y A083140 by number of 1's in binary expansion: A066884 (upward), A067576 (downward),
%Y A083140 by distance to next prime: A192179.
%Y A083140 Cf. A002110, A005867, A038110, A038111.
%K A083140 nonn,tabl,nice
%O A083140 2,1
%A A083140 _Yasutoshi Kohmoto_, Jun 05 2003
%E A083140 More terms from _Hugo Pfoertner_ and _Robert G. Wilson v_, Jun 13 2003