A285494

The On-Line Encyclopedia of Integer Sequences gets regularly updated with new integer sequences. One of the recent updates was contributed by me, A285494.

A285494 is the list of all numbers k so that its digit sum equals its number of distinct prime factors.
A number’s digit sum is the sum of all of its decimal digits. The number 62831853, for example, has a digit sum of 6+2+8+3+1+8+5+3 = 36.
A number’s number of distinct prime factors is the number of different prime numbers that multiply together to result in the original number. As an example, 62831853 = 3^2 \cdot 7 \cdot 127 \cdot 7853, so it has five prime factors of which four are distinct.
Thereby one can conclude that 62831853 is not an entry in this sequence, as 36 \neq 4.

The sequence is certainly infinite, as the number k = 2 \cdot 10^n with n \in \mathbb{N}^* has a digit sum of 2 + (0 \cdot n) = 2 and — because k = 2^{n+1} \cdot 5^n — exactly two distinct prime factors.

In the encyclopedia entry, I provided a Mathematica one-liner to compute the first few entries of this sequence. Since then, I have also written a Python two-liner to achieve the same goal.


(* Mathematica *)
Select[Range[2,10000],Total[IntegerDigits[#]]==Length[FactorInteger[#]]&]
Out = {20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010}
# Python 2.7
>>> def p(n):exec"i,f=2,set()\nwhile n>1:\n\tif n%i==0:f.add(i);n/=i;i=1\n\ti+=1";return len(f)
>>> print filter(lambda n:p(n)==sum(map(int,str(n))),range(2,10001))
[20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010]

JClock VIII

Interpreting the hour hand on a clock as a two-dimensional object on a plane, the hand’s tip can be seen as a complex number.
This clock converts the hour hand’s position into a complex number, sets the number’s length to the current minutes and displays it in the form a + b \cdot i.
The angle \phi is determined by the hours passed (\frac{2 \cdot \pi \cdot \text{hour}}{12} = \frac{\pi \cdot \text{hour}}{6}) but has to be slightly modified because a complex number starts at the horizontal axis and turns anti-clockwise whilst an hour hand starts at the vertical axis and turns — as the name implies — clockwise.
Thus \phi = (2 \cdot \pi - \frac{\pi \cdot \text{hour}}{6}) + \frac{\pi}{2} = (\frac{15 - \text{hour}}{6}) \cdot \pi.
The complex number’s length is simply determined by the minutes passed. Because the length must not be equal to 0, I simply add 1. |z| = k = \text{minute} + 1.
Lastly, to convert a complex number in the form k \cdot e^{\phi \cdot i} into the form a + b \cdot i, I use the formula k \cdot (\cos{\phi} + \sin{\phi} \cdot i) = a + b \cdot i.

17:04


# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 29th of July, 2016

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Triangular Squares

In a recent video Matt Parker showed a triangular number that also is a square number, 6, and asked if there were more.

A triangular number has the form \frac{n^2+n}{2} — shown by Euler — and a square number has the form m^2.
Triangular squares are those numbers for which \frac{n^2+n}{2} = m^2 with n,m \in \mathbb{N}.
Examples are \{0, 1, 6, 35, 204, 1189, 6930, \dots\} (sequence A001109 in OEIS).

To check if triangular numbers are square numbers is easy (code listed below), but a mathematical function would be nicer.
The first thing I tried was to define the triangular number’s square root as a whole number, \sqrt{\frac{n^2+n}{2}} = \lfloor \sqrt{\frac{n^2+n}{2}} \rfloor. This function does not return the square numbers that are triangular but the triangular numbers that are square.
The resulting sequence is \{0, 1, 8, 49, 288, 1681, 9800, \dots\} (sequence A001108 in OEIS).


# Python 2.7.7 Code
# Jonathan Frech 13th of July, 2016
#         edited 15th of July, 2016

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Circle Walk II

Continuing the idea from ‘Circle Walk’, I created ‘Circle Walk II’. In this program the entities get – based on their spawn time – a number (just to see the exact spawn time). As they spawn, they get put in a list. From this list their position around the center is calculated (in a similar way as in ‘Polygons’). Their distance to the center equals five times the number of entities (\text{distance to the center} = 5 \cdot \text{number of entities}), but cannot reach outside the screen. Their color is calculated based on their angle.

Usage

  • ‘Space’ to toggle if text is shown

The First ExampleThe Second ExampleThe Third Example


# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 15th of March, 2015
#         edited 30th of March, 2015
#     version II 1st  of April, 2015

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