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

2, 4, 3, 8, 9, 5, 16, 27, 25, 6, 32, 81, 125, 12, 7, 64, 243, 625, 18, 49, 10, 128, 729, 3125, 24, 343, 20, 11, 256, 2187, 15625, 36, 2401, 40, 121, 13, 512, 6561, 78125, 48, 16807, 50, 1331, 169, 14, 1024, 19683, 390625, 54, 117649, 80, 14641, 2197, 28, 15

Offset

1,1

Comments

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

See A284311 for further details.

Links

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Formula

From Antti Karttunen, Apr 17 2017: (Start)

A(n,1) = A005117(1+n), A(n,k) = A065642(A(n,k-1)). [A "dispersion" of A065642.]

A(A285329(n), A008479(n)) = n for all n >= 2.(End)

Example

Array starts:
    2    4     8     16      32      64      128
    3    9    27     81     243     729     2187
    5   25   125    625    3125   15625    78125
    6   12    18     24      36      48       54
    7   49   343   2401   16807  117649   823543
   10   20    40     50      80     100      160
   ...
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.

Mathematica

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

Prog

(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
(Scheme) (define (A284457 n) (A284311bi (A004736 n) (A002260 n))) ;; For A284311bi, see A284311. - Antti Karttunen, Apr 17 2017

Crossrefs

Cf. A284311, A005117, A007947, A065642, A182944.

Cf. A008479 (index of the column where n is located), A285329 (of the row).

Sequence in context: A304408 A048767 A269851 * A365656 A182944 A269385

Adjacent sequences: A284454 A284455 A284456 * A284458 A284459 A284460

Keyword

nonn,tabl

Author

Bob Selcoe, Mar 27 2017

Extensions

Edited by M. F. Hasler, Mar 27 2017

Status

approved