A102690 -id:A102690 - cp's OEIS frontend

A216653 Number A(n,k) of n-digit k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

10, 4, 90, 3, 6, 900, 2, 2, 22, 9000, 2, 2, 5, 68, 90000, 2, 1, 2, 12, 217, 900000, 2, 1, 1, 4, 25, 683, 9000000, 2, 0, 1, 3, 8, 53, 2163, 90000000, 2, 0, 1, 1, 3, 14, 116, 6837, 900000000, 2, 0, 1, 1, 2, 6, 25, 249, 21623, 9000000000

Offset: 1

Alois P. Heinz, Sep 12 2012

			A(1,1) = 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
A(1,2) = 4: 0, 1, 4, 9.
A(2,2) = 6: 16, 25, 36, 49, 64, 81.
A(3,3) = 5: 125, 216, 343, 512, 729.
A(4,4) = 4: 1296, 2401, 4096, 6561.
A(5,5) = 3: 16807, 32768, 59049.
A(6,6) = 3: 117649, 262144, 531441.
Square array A(n,k) begins:
:n\k|        1:     2:    3:   4:   5:  6:  7:  8
+---+--------------------------------------------
: 1 |       10,     4,    3,   2,   2,  2,  2,  2
: 2 |       90,     6,    2,   2,   1,  1,  0,  0
: 3 |      900,    22,    5,   2,   1,  1,  1,  1
: 4 |     9000,    68,   12,   4,   3,  1,  1,  1
: 5 |    90000,   217,   25,   8,   3,  2,  2,  1
: 6 |   900000,   683,   53,  14,   6,  3,  2,  1
: 7 |  9000000,  2163,  116,  25,  10,  5,  2,  2
: 8 | 90000000,  6837,  249,  43,  14,  7,  4,  2

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

Columns k=1-10 give: A063945, A062940, A062941, A102831, A216655, A216656, A216657, A216658, A216659, A216654.

Main diagonal gives: A102690.

r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, k) -r(10^(n-1), k) +`if`(n=1, 1, 0):
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);

A102691 Least n-expodigital number (i.e., numbers m such that m^n has exactly n decimal digits).

Original entry on oeis.org

0, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

Lekraj Beedassy, Jan 21 2005

10^(n-1) being the smallest n-digit number, n-expodigital numbers exist iff 10^(n-1) < 9^n, i.e., iff n-1 < n*log_10(9); this condition holds for all n up to 21 because beyond we have, for instance, 20 < 22*log_10(9) < 21. Thus numbers can be at most 21-expodigital.

			a(3)=5 because this is the first number followed by 6,7,8 and 9 which are all 3-expodigital: 5^3 = 125; 6^3 = 216; 7^3 = 343; 8^3 = 512; 9^3 = 729.

Cf. A102690.

Essentially the same as A067471. - R. J. Mathar, Aug 30 2008

a(n) = 10 - A102690(n).

Edited by Charles R Greathouse IV, Aug 03 2010

cp's OEIS Frontend

A216653 Number A(n,k) of n-digit k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A102691 Least n-expodigital number (i.e., numbers m such that m^n has exactly n decimal digits).

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