A120500 -id:A120500 - cp's OEIS frontend

A347040 Times on a 12-hour digital clock with 6 digits at which the three continuously moving hands of an analog clock, in the best approximation, enclose the same angles with one another, i.e., have the smallest sum of squares of the deviations from 120 degrees. When interpreting the terms as times of the day in the form hh:mm:ss, padding to the left with zeros is assumed.

2142, 4324, 12647, 14929, 23253, 25435, 33758, 35940, 44404, 50546, 54909, 61051, 65414, 71556, 80020, 82202, 90525, 92707, 101031, 103313, 111636, 113818, 122142, 124324

Offset: 1

Hugo Pfoertner, Aug 13 2021

An exact hit in which all angles are exactly 120 degrees is impossible. The smallest possible deviation occurs at two points in each display cycle, namely at 02:54:34.5617..., and at 09:05:25.4383... . With rounding to the nearest integer second, this corresponds to the terms a(6)=25435 and a(17)=90525.

The least squares clock solution actually occurs at an exact rational time, namely 5333364000/509173 seconds after 00:00:00, or 285998/509173 seconds after 02:54:34; and the exact least squares sum (in units of squared rotations) is 1/3055038 or (in units of squared clock-second-ticks) 3600/3055038 = 600/509173. - Robert B Fowler, Oct 29 2021

Henry Ernest Dudeney, Amusements in Mathematics, London, New York, Nelson, 1917. (Problem #63: The Stop-Watch)

Cf. A120500, A347039, A348637, A350141.

a(8) corrected by Robert B Fowler, Oct 29 2021

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

A348759 (43200/719)*{a(n)} are the times, measured in seconds from 00:00:00, at which the angle of the sector enclosing the three hands of an analog clock has a local minimum.

Original entry on oeis.org

0, 65, 131, 196, 261, 327, 392, 458, 523, 588, 654, 719

Offset: 1

Hugo Pfoertner, Oct 31 2021

The global minimum occurs at midnight or at noon, when the positions of all three clock hands coincide exactly. All other minima are characterized by the fact that the position of the second hand coincides exactly with the position of the hour hand. The neighboring situation in which the second hand coincides with the minute hand leads in all possible cases to a slightly larger angle between the minute and hour hand.

In order to determine the minimum central angle of the sector between the hour hand and the minute hand, the configurations located next to the exact matches occurring 11 times per clock cycle (corresponding to A120500) must be considered, in which the position of the second hand is either equal to the minute hand or is equal to the hour hand.
The position of the second hand coincides 719 times per clock cycle with the hour hand and 12*59 = 708 times with the minute hand. The following coincidences lie in the vicinity of the first local minimum, which clearly occurs shortly after 01:05.
Coincidence of minute hand and second hand after 64/59 hours (= 230400/59 seconds). At this point in time, the minute hand has an angle of 2*Pi*5/59 = Pi*10/59 radians. The hour hand is at 2*Pi*(230400/59)/43200 = Pi*32/177 radians. The difference angle is therefore Pi * 2/177 radians = 2.0338983 degrees.
Coincidence of hour hand and second hand after 2808000/719 seconds. This corresponds with an angle of Pi*130/719 radians. At the same time, the minute hand is at Pi*122/719 radians, making a difference angle of Pi*8/719 radians = 2.002781641 degrees.
This angle is slightly smaller than the aforementioned difference angle of 2.0338983 degrees, i.e., the coincidence between the hour hand and the second hand determines the exact position of the minimum and thus a(2) = 2808000/43200 = 65.
Rounded to full seconds, the two points in time that are approximately 0.34 seconds apart lead to the same full seconds value of 5 s, which results in A348758(2) = 10505 corresponding to the time of 01:05:05.
A similar calculation for the other candidate configurations shows that the coincidence between the second hand and the hour hand always leads to the smaller difference angle. The terms of A348758 can easily be determined by converting the fractional second values of the present sequence to the hh:mm:ss form, including rounding to the nearest second and then dropping the two colons.

Cf. A120500, A347040, A348758.

a(n) = round(n*719/11). - Robert B Fowler, Dec 26 2022

A335789 a(n) = time to the nearest second at the n-th instant (n>=0) when the hour and minute hands on a clock face coincide, starting at time 0:00.

Original entry on oeis.org

0, 3927, 7855, 11782, 15709, 19636, 23564, 27491, 31418, 35345, 39273, 43200, 47127, 51055, 54982, 58909, 62836, 66764, 70691, 74618, 78545, 82473, 86400, 90327, 94255, 98182, 102109, 106036, 109964, 113891, 117818, 121745, 125673, 129600, 133527, 137455, 141382

Offset: 0

Sean Lestrange, Aug 14 2020

nonn

After 12 hours or 43200 seconds, the hands overlap at 12:00 and the cycle repeats.

			For n=1, 3927 would correspond to slightly after 01:05.

Mathlete's Corner, How many times during the day do the hands of a clock overlap?
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A120500 (as HHMMSS).

L=[]
n=0
while n<50:
    L.append(round(numerical_approx((n+n/11))*3600))
    n+=1
print(L)

a(n) = round(n*43200/11).

A347039 Times on the display of a 24-hour digital clock with 6 digits, rounded to full seconds, at which the hour and minute hands of an analog clock form a right angle. Terms with fewer than 6 digits are to be assumed filled with zeros to the left.

Original entry on oeis.org

1622, 4905, 12149, 15433, 22716, 30000, 33244, 40527, 43811, 51055, 54338, 61622, 64905, 72149, 75433, 82716, 90000, 93244, 100527, 103811, 111055, 114338, 121622, 124905, 132149, 135433, 142716, 150000, 153244, 160527, 163811, 171055, 174338, 181622, 184905, 192149, 195433, 202716, 210000, 213244, 220527, 223811, 231055, 234338

Offset: 1

Hugo Pfoertner, Aug 13 2021

Configurations in which the hour and minute hands of an analog clock enclose a right angle occur 44 times each day. The first occurrence happens at 00:16:22. The period is 1963 + 7/11 seconds, i.e., 32 minutes 43.6363... seconds.

Cf. A120500, A335789.

A378864 Decimal expansion of 11*Pi/360.

Original entry on oeis.org

0, 9, 5, 9, 9, 3, 1, 0, 8, 8, 5, 9, 6, 8, 8, 1, 2, 6, 7, 3, 0, 8, 0, 2, 9, 9, 2, 2, 6, 6, 8, 7, 3, 6, 9, 9, 2, 3, 9, 3, 5, 7, 9, 5, 3, 8, 6, 9, 7, 9, 4, 9, 0, 0, 0, 8, 5, 3, 4, 5, 5, 2, 9, 2, 0, 9, 4, 0, 5, 5, 0, 1, 3, 0, 3, 1, 8, 9, 7, 1, 9, 4, 0, 2, 5, 2, 3, 2, 8, 6, 3, 2, 9, 8, 9, 8, 0, 2, 1, 5, 2, 1, 6, 7, 6, 7, 4, 7

Offset: 0

Gonzalo Martínez, Dec 09 2024

The smallest angle, measured in radians, formed by the hour hand and the minute hand at 00:01 hours.
When 1 minute has passed after 00:00, the minute hand moves 6 degrees, while the hour hand moves 0.5 degrees, so the smallest angle they form is 5.5 degrees, which in radians is 11*Pi/360.
Meanwhile, the largest angle formed by the hands is 709*Pi/360 radians.

			0.0959931088596881267308029922668736992393579538697949...

Index entries for transcendental numbers

Cf. A019685, A120500, A348758.

Mathematica
```
RealDigits[11*Pi/360, 10, 111][[1]]
```
PARI
```
11*Pi/360 \\ Amiram Eldar, Dec 10 2024
```

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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.

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A348759 (43200/719)*{a(n)} are the times, measured in seconds from 00:00:00, at which the angle of the sector enclosing the three hands of an analog clock has a local minimum.

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A335789 a(n) = time to the nearest second at the n-th instant (n>=0) when the hour and minute hands on a clock face coincide, starting at time 0:00.

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A347039 Times on the display of a 24-hour digital clock with 6 digits, rounded to full seconds, at which the hour and minute hands of an analog clock form a right angle. Terms with fewer than 6 digits are to be assumed filled with zeros to the left.

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A378864 Decimal expansion of 11*Pi/360.

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