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A305932 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 2^k has n digits '0' (conjectured).

%I A305932 #23 Aug 30 2023 07:28:07
%S A305932 0,1,2,3,4,5,6,7,8,9,13,14,15,16,18,19,24,25,27,28,31,32,33,34,35,36,
%T A305932 37,39,49,51,67,72,76,77,81,86,10,11,12,17,20,21,22,23,26,29,30,38,40,
%U A305932 41,44,45,46,47,48,50,57,58,65,66,68,71,73,74,75,84,85,95,96,122,124,129,130,149,151,184,43,53,61,69,70
%N A305932 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 2^k has n digits '0' (conjectured).
%C A305932 A partition of the nonnegative integers (the rows being the subsets).
%C A305932 Although it remains an open problem to provide a proof that the rows are complete (as are all terms of A020665), we can assume it for the purpose of this sequence.
%C A305932 Read as a flattened sequence, a permutation of the nonnegative integers.
%H A305932 M. F. Hasler, <a href="https://oeis.org/wiki/Zeroless_powers">Zeroless powers</a>, OEIS Wiki, March 2014.
%F A305932 Row n = { k >= 0 | A027870(k) = n }.
%e A305932 The table reads:
%e A305932 n \ k's
%e A305932 0 : 0, 1, ..., 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, (...), 81, 86 (cf. A007377)
%e A305932 1 : 10, 11, 12, 17, 20, 21, 22, 23, 26, 29, 30, 38, 40, 41, 44, (...), 151, 184
%e A305932 2 : 42, 52, 54, 55, 56, 59, 60, 62, 63, 64, 78, 80, 82, 92, 107, (...), 171, 231
%e A305932 3 : 43, 53, 61, 69, 70, 83, 87, 89, 90, 93, 109, 112, 114, 115, (...), 221, 359
%e A305932 4 : 79, 91, 94, 97, 106, 118, 126, 127, 137, 139, 157, 159, 170, (...), 241, 283
%e A305932 5 : 88, 98, 99, 103, 104, 113, 120, 143, 144, 146, 152, 158, 160, (...), 343, 357
%e A305932 ...
%e A305932 Column 0 is A031146: least k such that 2^k has n digits '0' in base 10.
%e A305932 Row lengths = number of powers of 2 with exactly n '0's = (36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, 35, 29, 27, 27, 39, 34, 34, 28, 29, ...): not in the OEIS.
%e A305932 Largest number in row n = (86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, 696, 824, 634, 732, 890, 895, 848, 823, 929, 1092, ...): not in the OEIS.
%e A305932 Row number of n = Number of '0's in 2^n = A027870: (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, ...).
%e A305932 Inverse permutation (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 10, 11, 12, 13, 39, 14, 15, 40, 41, 42, 43, 16, 17, 44, 18, 19, 45, 46, 20, 21, ...) is not in the OEIS.
%t A305932 mx = 1000; g[n_] := g[n] = DigitCount[2^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 A305932 (PARI) apply( A305932_row(n,M=200*(n+1))=select(k->A027870(k)==n,[0..M]), [0..20]) \\ A027870(k)=#select(d->!d, digits(2^k))
%Y A305932 Cf. A007377, A031146.
%Y A305932 Sequence A027870 yields the row number of a given integer.
%Y A305932 Cf. A305933 (analog for 3^n), A305924 (for 4^n), ..., A305929 (for 9^n).
%K A305932 nonn,base,tabf,nice
%O A305932 0,3
%A A305932 _M. F. Hasler_, Jun 14 2018