A305932 - OEIS

%I #23 Aug 30 2023 07:28:07

%S 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 37,39,49,51,67,72,76,77,81,86,10,11,12,17,20,21,22,23,26,29,30,38,40,

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

%C A partition of the nonnegative integers (the rows being the subsets).

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

%H M. F. Hasler, <a href="https://oeis.org/wiki/Zeroless_powers">Zeroless powers</a>, OEIS Wiki, March 2014.

%F Row n = { k >= 0 | A027870(k) = n }.

%e The table reads:

%e n \ k's

%e 0 : 0, 1, ..., 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, (...), 81, 86 (cf. A007377)

%e 1 : 10, 11, 12, 17, 20, 21, 22, 23, 26, 29, 30, 38, 40, 41, 44, (...), 151, 184

%e 2 : 42, 52, 54, 55, 56, 59, 60, 62, 63, 64, 78, 80, 82, 92, 107, (...), 171, 231

%e 3 : 43, 53, 61, 69, 70, 83, 87, 89, 90, 93, 109, 112, 114, 115, (...), 221, 359

%e 4 : 79, 91, 94, 97, 106, 118, 126, 127, 137, 139, 157, 159, 170, (...), 241, 283

%e 5 : 88, 98, 99, 103, 104, 113, 120, 143, 144, 146, 152, 158, 160, (...), 343, 357

%e ...

%e Column 0 is A031146: least k such that 2^k has n digits '0' in base 10.

%e 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 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 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 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 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 (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 Cf. A007377, A031146.

%Y Sequence A027870 yields the row number of a given integer.

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

%K nonn,base,tabf,nice

%O 0,3

%A _M. F. Hasler_, Jun 14 2018