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A284457 Square array whose rows list numbers with the same squarefree kernel (A007947): Transpose of A284311.

%I A284457 #22 Apr 17 2017 22:43:20
%S A284457 2,4,3,8,9,5,16,27,25,6,32,81,125,12,7,64,243,625,18,49,10,128,729,
%T A284457 3125,24,343,20,11,256,2187,15625,36,2401,40,121,13,512,6561,78125,48,
%U A284457 16807,50,1331,169,14,1024,19683,390625,54,117649,80,14641,2197,28,15
%N A284457 Square array whose rows list numbers with the same squarefree kernel (A007947): Transpose of A284311.
%C A284457 The first column contains the squarefree numbers A005117; each row lists all numbers having the same prime divisors. If T[m,1] is prime then the row contains the powers of that prime. Yields A182944 if only these rows with prime powers (A000961) are kept. - _M. F. Hasler_, Mar 27 2017
%C A284457 See A284311 for further details.
%H A284457 Alois P. Heinz, <a href="/A284457/b284457.txt">Antidiagonals n = 1..141, flattened</a>
%F A284457 From _Antti Karttunen_, Apr 17 2017: (Start)
%F A284457 A(n,1) = A005117(1+n), A(n,k) = A065642(A(n,k-1)). [A "dispersion" of A065642.]
%F A284457 A(A285329(n), A008479(n)) = n for all n >= 2.(End)
%e A284457 Array starts:
%e A284457     2    4     8     16      32      64      128
%e A284457     3    9    27     81     243     729     2187
%e A284457     5   25   125    625    3125   15625    78125
%e A284457     6   12    18     24      36      48       54
%e A284457     7   49   343   2401   16807  117649   823543
%e A284457    10   20    40     50      80     100      160
%e A284457    ...
%e A284457 Row 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 A284457 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, 1, -1}] // Flatten
%o A284457 (PARI) A284457(m,n)={for(a=2,m^2+1,(core(a)!=a||m--)&&next;m=factor(a)[,1]; for(k=1,9e9,factor(k*a)[,1]==m&&!n--&&return(k*a)))} \\ _M. F. Hasler_, Mar 27 2017
%o A284457 (Scheme) (define (A284457 n) (A284311bi (A004736 n) (A002260 n))) ;; For A284311bi, see A284311. - _Antti Karttunen_, Apr 17 2017
%Y A284457 Cf. A284311, A005117, A007947, A065642, A182944.
%Y A284457 Cf. A008479 (index of the column where n is located), A285329 (of the row).
%K A284457 nonn,tabl
%O A284457 1,1
%A A284457 _Bob Selcoe_, Mar 27 2017
%E A284457 Edited by _M. F. Hasler_, Mar 27 2017