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%I #43 Apr 17 2017 22:43:11
%S 2,3,4,5,9,8,6,25,27,16,7,12,125,81,32,10,49,18,625,243,64,11,20,343,
%T 24,3125,729,128,13,121,40,2401,36,15625,2187,256,14,169,1331,50,
%U 16807,48,78125,6561,512,15,28,2197,14641,80,117649,54,390625,19683,1024
%N Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).
%C A permutation of the natural numbers > 1.
%C T(1,k)= A005117(m) with m > 1; terms in column k = T(1,k) * A162306(T(1,k)) only not bounded by T(1,k). Let T(1,k) = b. Terms in column k are multiples of b and numbers c such that c | b^e with e >= 0. Alternatively, terms in column k are multiples bc with c those numbers whose prime divisors p also divide b. - _Michael De Vlieger_, Mar 25 2017
%H Alois P. Heinz, <a href="/A284311/b284311.txt">Antidiagonals n = 1..141, flattened</a>
%F From _Antti Karttunen_, Apr 17 2017: (Start)
%F A(1,k) = A005117(1+k), A(n,k) = A065642(A(n-1,k)).
%F A(A008479(n), A285329(n)) = n for all n >= 2.
%F (End)
%e Array starts:
%e 2 3 5 6 7 10 11 13 14 15
%e 4 9 25 12 49 20 121 169 28 45
%e 8 27 125 18 343 40 1331 2197 56 75
%e 16 81 625 24 2401 50 14641 371293 98 135
%e 32 243 3125 36 16807 80 161051 4826809 112 225
%e 64 729 15625 48 117649 100 1771561 62748517 196 375
%e 128 2187 78125 54 823543 160 19487171 815730721 224 405
%e Column 6 is: T(1,6) = 2*5; T(2,6) = 2^2*5; T(3,6) = 2^3*5; T(4,6) = 2*5^2; T(5,6) = 2^4*5, etc.
%t f[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sf}, sf = n + sgn; While[c < Abs[k], While[! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[sgn < 0, sf--, sf++]; c++]; sf + If[sgn < 0, 1, -1]] (* after _Robert G. Wilson v_ at A005117 *); T[n_, k_] := T[n, k] = Which[And[n == 1, k == 1], 2, k == 1, f@ T[n - 1, k], PrimeQ@ T[n, 1], T[n, 1]^k, True, Module[{j = T[n, k - 1]/T[n, 1] + 1}, While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; Table[T[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* _Michael De Vlieger_, Mar 25 2017 *)
%o (Scheme)
%o (define (A284311 n) (A284311bi (A002260 n) (A004736 n)))
%o (define (A284311bi row col) (if (= 1 row) (A005117 (+ 1 col)) (A065642 (A284311bi (- row 1) col))))
%o ;; _Antti Karttunen_, Apr 17 2017
%Y Cf. A005117 (squarefree numbers), A033845 (column 4), columns 1,2,3,5 are powers of primes, A033846 (column 6), A033847 (column 9), A033849 (column 10).
%Y The columns that are powers of primes have indices A071403(n) - 1. - _Michel Marcus_, Mar 24 2017
%Y See also A007947; the k-th column of the array corresponds to the numbers with radical A005117(k+1). - _Rémy Sigrist_, Mar 24 2017
%Y Cf. A284457 (this sequence read by antidiagonals upwards), A285321 (a similar array, but columns come in different order).
%Y Cf. A065642.
%Y Cf. A008479 (index of the row where n is located), A285329 (of the column).
%K nonn,tabl
%O 1,1
%A _Bob Selcoe_, Mar 24 2017
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