Your task is to generate the "Primitive and Basic" numbers, which are made like so:

Start with 110. Take the digits of its base-1 representation (110 = 11) and interpret them as base-2 digits. This gives 12 = 110. Now, add the second prime number – 310 – to the result. This will give you 410, which is our first "Primitive and Basic" (PB) number, or PB(1) (you can now see where I got the name from).

Next, take 410 = 1002 and interpret the digits of its base-2 representation as base-3 digits. This gives 1003 = 910. Add the third prime number – 510 – to the result, which gives 1410. That's the second PB number, or PB(2).

You can see where this is going: take 1410 = 1123 and interpret the digits of its base-3 representation as base-4 digits. This gives 1124 = 2210. Add the fourth prime number – 710 – to the result, which gives 2910.

This continues on and on forever.

In general, PB(n) is equal to PB(n-1), converted to base n and from base n+1 to integer, plus the (n+1)th prime.

The first few terms of this sequence are:

4, 14, 29, 52, 87, 132, 185...

You must write a function or program that creates these numbers. It must take a positive integer n as input, and return PB(n).


1 -> 4
5 -> 87

Rules and specs:

  • Your program must be able to handle all PB numbers below 231-1.
  • Standard loopholes apply.
  • The input may be 0-indexed or 1-indexed, whichever you prefer.
  • The input is guaranteed to be a valid index.
  • The output must be in base 10.

This is , so shortest code in bytes wins!

Unfortunately, there isn't an OEIS sequence for this. You can propose to OEIS to add this sequence in if you want.

  • \$\begingroup\$ I find "turn it into base 3" misleading. That sounds like I should convert either 4 or 100 to base 3 which is either 11 or 10201. What you seem to intend is that the representation from the previous base should now be treated as digits in the next base. \$\endgroup\$ Commented Dec 20, 2016 at 9:26
  • \$\begingroup\$ What are we supposed to do? Create a program/function takes takes input and gives the nth PB term? Or are we supposed to generate all terms below 2^31-1? \$\endgroup\$
    – user41805
    Commented Dec 20, 2016 at 9:28
  • \$\begingroup\$ Sorry for the misleading thing: your program must be able to handle terms that are up to 2^31-1. It's meant to take an input n and return the output PB(n). \$\endgroup\$
    – clismique
    Commented Dec 20, 2016 at 9:29
  • \$\begingroup\$ I just hope all these comments get me the "Blue in the Face" hat \$\endgroup\$
    – user41805
    Commented Dec 20, 2016 at 9:30
  • 1
    \$\begingroup\$ I didn't understand the task the first time I read it, so I rephrased it in an attempt to make it clearer. Please feel free to roll back any changes you disagree with. \$\endgroup\$
    – Dennis
    Commented Dec 21, 2016 at 7:01

4 Answers 4


Ruby, 126 bytes

(r.unshift o%i;o/=i)while o>0

Jelly, 13 bytes


Try it online!

How it works

’ß1¹?bḅ‘+‘ÆN$  Main link. Argument: n

’              Decrement; yield n-1.
   ¹?          If n-1 is non-zero:
 ß               Recursively call the main link with argument n-1.
  1              Yield 1.
     b         Convert the result to base n.
      ḅ‘       Convert from base n+1 to integer.
         ‘ÆN$  Increment n and yield the (n+1)-th prime.
        +      Add the results to both sides.

Mathematica, 66 bytes


Unnamed function taking a single (1-indexed) integer argument and returning an integer.

Fold[...,4,2~Range~#+1] iterates a function of two arguments, with the first argument beginning at 4 and thereafter set to the output of the previous iteration, while the second argument iterates over the set {3,4,...,#+1} (where # is the original integer argument).

The function in question is IntegerDigits[#,#2-1]~FromDigits~#2+Prime@#2. First, IntegerDigits[#,#2-1] converts the first argument into a list of its digits when written in the previous base; then, ...~FromDigits~#2 converts that list of digits back into an integer, assuming the digits are now in the current base. Finally, the #2th prime is added to the result.


Perl 6, 95 bytes

{my $b=1;(4,{grep(*.is-prime,2..*)[++$b]+.polymod($b xx*).&{sum $_ Z*($b+1 X**0..*)}}...*)[$_]}

Zero-based indexed.


Since the language's built-in .base and .parse-base methods are string-based, they only work up to base 36. So in order to fulfill the task requirement that large inputs must be supported, the base conversion is done as follows:

  • .polymod($b xx *): Convert the current number to a list representing its digits in base $b.
  • sum $_ Z* ($b+1 X** 0..*): Interpret the resulting digits as a number in base $b+1.

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