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A292579 Inverse to A055643: Consider n as a number written in base 60, using 2 decimal digits for each base-60 digit (which we allow to be in the range 0 .. 99), and write n in base 10.

%I A292579 #31 Oct 23 2022 01:25:23
%S A292579 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
%T A292579 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,
%U A292579 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,60,61,62,63,64,65,66,67,68,69
%N A292579 Inverse to A055643: Consider n as a number written in base 60, using 2 decimal digits for each base-60 digit (which we allow to be in the range 0 .. 99), and write n in base 10.
%C A292579 For the first 240000 terms, also the conversion of a digital clock display H:MM:SS to seconds, i.e., a(HMMSS) yields the number of seconds after midnight corresponding to that time, with leading zeros allowed: 0 <= H <= 24, 0 <= MM, SS <= 59.
%C A292579 Numbers where SS or MM etc. are > 59 are strictly spoken invalid in base 60. However, in order to "fill in the gaps" in a meaningful way, we simply do allow SS, MM, etc. to take values up to 99. (The ISO specification for writing time and date does actually allow SS to take values including 60 in order to account for leap seconds, as it allows H to be equal to 24 to denote midnight at the end of a day.) Therefore a(60+k) = 60+k for k < 40, but a(100+k) = 60+k, a(200+k) = 120+k etc, for 0 <= k <= 99.
%F A292579 a(n) = n - [n/100]*40 - [n/10^4]*40*60 - [n\10^6]*40*60^2 - ..., where [.] denotes the floor function.
%F A292579 A292579 o A055643 = A001477 (the identity map on the set of nonnegative integers), i.e., for all n >= 0, A292579(A055643(n)) = n.
%F A292579 Also, A055643(A292579(n)) = n for all n in the range of A055643, which are the "valid base-60 numbers", i.e., excluding 60, ..., 99, 160, ..., 199 etc. The latter (60, ...) are "converted" by A055643 o A292579 to correct base-60 numbers, i.e., 60 -> 100, 99 = 60 + 39 -> 139, 9999 -> 14039 since 99*60 + 99 = 6039 = 1*60^2 + 40*60 + 39 = 14039[60].
%e A292579 Interpreted in base-60, HMMSS[60] = H*60^2 + MM*60 + SS. Thus, e.g., a(100) = 60, a(123456) = 12*3600 + 34*60 + 56 = 45296.
%e A292579 In contrast to (most) digital clocks, MM and SS may take values up to 99. For example, a(199) = 1*60 + 99 = 159.
%o A292579 (PARI) A292579(n)=fromdigits(digits(n,100),60)
%Y A292579 Cf. A055643.
%K A292579 nonn,base
%O A292579 0,3
%A A292579 _M. F. Hasler_, Jan 09 2018