A305925 - OEIS

%I #22 Jun 23 2018 07:24:19

%S 0,1,2,3,4,5,6,7,9,10,11,17,18,30,33,58,8,12,14,15,16,19,20,21,22,25,

%T 26,31,41,42,43,85,13,23,24,27,28,29,32,36,37,56,57,107,34,35,38,39,

%U 50,54,59,74,75,84,112,40,44,46,47,49,51,60,73,78,79,82,83,86,88,89,95,96,97,106,113,127

%N Irregular table read by rows in which row n >= 0 lists all k >= 0 such that the decimal representation of 5^k has n digits '0' (conjectured).

%C The set of (nonempty) rows is a partition of the nonnegative integers.

%C Read as a flattened sequence, a permutation of the nonnegative integers.

%C In the same way, another choice of (basis, digit, base) = (m, d, b) different from (5, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.

%C It remains an open problem to provide a proof that the rows are complete, in the same way as each of the terms of A020665 is unproved.

%C We can also decide that the rows are to be truncated as soon as no term is found within a sufficiently large search limit. (For all of the displayed rows, there is no additional term up to many orders of magnitude beyond the last term.) That way the rows are well-defined, but we are no more guaranteed to get a partition of the integers.

%C The author finds this sequence "nice", i.e., appealing (as well as, e.g., the variant A305933 for basis 3) in view of the idea of partitioning the integers in such an elementary yet highly nontrivial way, and the remarkable fact that the rows are just roughly one line long. Will this property remain for large n, or else, how will the row lengths evolve?

%e The table reads:

%e n \ k's

%e 0 : 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58 (cf. A008839)

%e 1 : 8, 12, 14, 15, 16, 19, 20, 21, 22, 25, 26, 31, 41, 42, 43, 85

%e 2 : 13, 23, 24, 27, 28, 29, 32, 36, 37, 56, 57, 107

%e 3 : 34, 35, 38, 39, 50, 54, 59, 74, 75, 84, 112

%e 4 : 40, 44, 46, 47, 49, 51, 60, 73, 78, 79, 82, 83, 86, 88, 89, 95, 96, 97, 106, 113, 127

%e 5 : 48, 55, 61, 67, 77, 91, 102, 110, 111, 126, 148, 157

%e ...

%e The first column is A063585: least k such that 5^k has n digits '0' in base 10.

%e Row lengths are 16, 16, 12, 11, 21, 12, 17, 14, 16, 17, 14, 13, 16, 18, 13, 14, 10, 10, 21, 7,... (not in OEIS).

%e Last terms of the rows are (58, 85, 107, 112, 127, 157, 155, 194, 198, 238, 323, 237, 218, 301, 303, 324, 339, 476, 321, 284, ...), not in OEIS.

%e The inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 16, 8, 9, 10, 17, 32, 18, 19, 20, 11, 12, 21, 22, 23, 24, 33, 34, 25, 26, 35, 36, 37, 13, 27,...), not in OEIS.

%t mx = 1000; g[n_] := g[n] = DigitCount[5^n, 10, 0]; f[n_] := Select[Range@mx, g@# == n &]; Table[f@n, {n, 0, 4}] // Flatten (* _Robert G. Wilson v_, Jun 20 2018 *)

%o (PARI) apply( A305925_row(n,M=60*(n+1))=select(k->#select(d->!d,digits(5^k))==n,[0..M]), [0..19])

%Y Cf. A008839, A063585.

%Y Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).

%K nonn,base,tabf

%O 0,3

%A _M. F. Hasler_, Jun 19 2018