A049345 n written in primorial base.

0, 1, 10, 11, 20, 21, 100, 101, 110, 111, 120, 121, 200, 201, 210, 211, 220, 221, 300, 301, 310, 311, 320, 321, 400, 401, 410, 411, 420, 421, 1000, 1001, 1010, 1011, 1020, 1021, 1100, 1101, 1110, 1111, 1120, 1121, 1200, 1201, 1210, 1211, 1220, 1221, 1300, 1301, 1310, 1311

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• base
• easy
• nice

Comments

Places reading from right have values (1, 2, 6, 30, 210, ...) = primorials.

For n < 10 * 7# = 2100: a(n) = concatenation of n-th row in A235168 and for n > 0: A055642(a(n)) = A235224(n); for larger numbers the representation in A235168 is more appropriate. - Reinhard Zumkeller, Jan 05 2014

In the long run, numbers have fewer digits in the primorial base than in the factorial base (cf. A007623), since factorial(n) < n^n < primorial(n) for n > 12. However, the point where the digits become larger than 9 comes earlier: as soon as 10*7*5*3*2 = 2100 for the primorial base vs 10! = 3628800 in the factorial base. From there on, the representation using concatenation of digits written in decimal becomes ambiguous. - M. F. Hasler, Sep 22 2014

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..2099
• Anthony Overmars, Survey of RSA Vulnerabilities, in: Menachem Domb (ed.), Modern Cryptography - Current Challenges and Solutions, Intechopen, 2019, pp. 17-41. See pp. 29-30.
• Index entries for sequences related to primorial base.

Crossrefs

Cf. A000040, A002110 (primorials), A235168, A235224, A276086, A276150.

Cf. factorial base A007623.

Programs

Haskell

a049345 n | n < 2100  = read $ concatMap show (a235168_row n) :: Int
          | otherwise = error "ambiguous primorial representation"
-- Reinhard Zumkeller, Jan 05 2014

Mathematica

Table[FromDigits@ IntegerDigits[n, MixedRadix[Reverse@ Prime@ Range@ 8]], {n, 0, 51}] (* Michael De Vlieger, Aug 23 2016, Version 10.2 *)

PARI

A049345(n, p=2) = if(nA049345(n\p, nextprime(p+1))*10 + n%p) \\ Valid at least up to the point where digits > 9 would arise (n=10*7*5*3*2), thereafter the definition of the sequence is ambiguous. M. F. Hasler, Sep 22 2014

Python

from sympy import nextprime
def a(n, p=2):
    if n>2099: print("Error! Ambiguous primorial representation when n is larger than 2099")
    else: return n if n
Indranil Ghosh, Jun 22 2017

Scheme

(define (A049345 n) (if (>= n 2100) (error "A049345: ambiguous primorial representation when n is larger than 2099:" n) (let loop ((n n) (s 0) (t 1) (i 1)) (if (zero? n) s (let* ((p (A000040 i)) (d (modulo n p))) (loop (/ (- n d) p) (+ (* t d) s) (* 10 t) (+ 1 i)))))))
;; Antti Karttunen, Aug 26 2016