A348759 -id:A348759 - cp's OEIS frontend

A120500 Times in hours,minutes and seconds (to the nearest second) at which the smoothly crossing minute and hour hands of an analog clock coincide, over a period of one complete 12-hour sweep of the hour hand.

0, 10527, 21055, 31622, 42149, 52716, 63244, 73811, 84338, 94905, 105433, 120000

Offset: 0

Lekraj Beedassy, Aug 06 2006

A subsequence of A121577. - Lekraj Beedassy, Aug 09 2006

			52716, for instance, in the sequence is meant to be read 5:27:16 or 5hr27mn16s.
We have a(3)=round(43200*3/11) to base 60(double-spaced), i.e., 11782=3*60^2 +16*60 + 22*1 to base 60, which is 31622.

M. Gardner, Science Fiction Puzzle Tales, Problem 28 pp. 90;141 Clarkson N. Potter NY 1981.
M. Gardner, Mathematical Puzzles of Sam Loyd, Problem 43 pp. 40;137 Dover NY 1959.
A. Jouette, Le Secret Des Nombres, Problem 52 pp. 176;269 Albin Michel Paris 1996.

T. Eveilleau, Les aiguilles de l'horloge (in French).

Cf. A348758, A348759.

fix[{a_,b_,c_}]:=FromDigits[Flatten[{a,PadLeft[IntegerDigits[b], 2], PadLeft[ IntegerDigits[c],2]}]]; Join[{0},fix/@Table[ IntegerDigits[ Round[(43200n)/11],60],{n,11}]] (* Harvey P. Dale, Oct 05 2017 *)

a(n)=round(43200*n/11) expressed in double-spaced sexagesimal scale. In other words, the hour and minute hands line up at 11 successive positions after every (12/11)hr, i.e., 1hr5min27s and 3/11s from noon or midnight.

A348758 Times on a 12-hour digital clock with 6 digits at which the angle of the sector enclosing the three continuously moving hands of an analog clock has a local minimum.

Original entry on oeis.org

0, 10505, 21111, 31616, 42122, 52727, 63233, 73838, 84344, 94849, 105455, 120000

Offset: 1

Hugo Pfoertner, Oct 31 2021

The inclusion of the second hand leads to solutions differing from those of A120500. Apart from the perfect match at midnight or noon, the other minima are characterized by small angles > 0 between the hour and the minute hands and a coincidence of the second hand with one of the other two hands. It turns out that the coincidence of the minute and second hands never leads to the smaller angle value. The exact times in seconds are given by A348759, with rounding to nearest second applied to determine the terms of this sequence.

			See A348759.

Cf. A120500, A347040, A348759.

vector(12, i, my(t=((i-1)*719\/11)*43200\/719); t\3600*10000 + t%3600\60*100 + t%60) \\ Ruud H.G. van Tol, Jan 07 2025

a(n) = round(A348759(n)*43200/719) seconds since 00:00:00, expressed as hhmmss. - Robert B Fowler, Jan 05 2023

cp's OEIS Frontend

A120500 Times in hours,minutes and seconds (to the nearest second) at which the smoothly crossing minute and hour hands of an analog clock coincide, over a period of one complete 12-hour sweep of the hour hand.

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