A285494 (#171)
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, , 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 .
The sequence is certainly infinite, as the number with
has a digit sum of
and — because
— 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]