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A305925 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).

%I A305925 #26 Mar 14 2025 15:51:52
%S A305925 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 A305925 26,31,41,42,43,85,13,23,24,27,28,29,32,36,37,56,57,107,34,35,38,39,
%U A305925 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 A305925 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 A305925 The set of (nonempty) rows is a partition of the nonnegative integers.
%C A305925 Read as a flattened sequence, a permutation of the nonnegative integers.
%C A305925 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 A305925 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 A305925 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 A305925 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 A305925 The table reads:
%e A305925 n \ k's
%e A305925 0 : 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58 (cf. A008839)
%e A305925 1 : 8, 12, 14, 15, 16, 19, 20, 21, 22, 25, 26, 31, 41, 42, 43, 85
%e A305925 2 : 13, 23, 24, 27, 28, 29, 32, 36, 37, 56, 57, 107
%e A305925 3 : 34, 35, 38, 39, 50, 54, 59, 74, 75, 84, 112
%e A305925 4 : 40, 44, 46, 47, 49, 51, 60, 73, 78, 79, 82, 83, 86, 88, 89, 95, 96, 97, 106, 113, 127
%e A305925 5 : 48, 55, 61, 67, 77, 91, 102, 110, 111, 126, 148, 157
%e A305925 ...
%e A305925 The first column is A063585: least k such that 5^k has n digits '0' in base 10.
%e A305925 Row lengths are 16, 16, 12, 11, 21, 12, 17, 14, 16, 17, 14, 13, 16, 18, 13, 14, 10, 10, 21, 7,... (A305945).
%e A305925 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, ...), A306115.
%e A305925 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 A305925 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 A305925 (PARI) apply( A305925_row(n,M=60*(n+1))=select(k->#select(d->!d,digits(5^k))==n,[0..M]), [0..19])
%Y A305925 Cf. A008839, A063585.
%Y A305925 Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
%K A305925 nonn,base,tabf
%O A305925 0,3
%A A305925 _M. F. Hasler_, Jun 19 2018