A235168 -id:A235168 - cp's OEIS frontend

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

R. K. Guy

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

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.

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

Cf. factorial base A007623.

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

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

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

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 nIndranil Ghosh, Jun 22 2017

(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

A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 4

Offset: 0

Antti Karttunen, Aug 22 2016

The sum of digits of n in primorial base is odd if n is 1 or 2 (mod 4) and even if n is 0 or 3 (mod 4). Proof: primorials are 1 or 2 (mod 4) and a(n) can be constructed via the greedy algorithm. So if n = 4k + r where 0 <= r < 4, 4k needs an even number of primorials and r needs hammingweight(r) = A000120(r) primorials. Q.E.D. - David A. Corneth, Feb 27 2019

			For n=24, which is "400" in primorial base (as 24 = 4*(3*2*1) + 0*(2*1) + 0*1, see A049345), the sum of digits is 4, thus a(24) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A000120, A001222, A002110, A049345, A053589, A235168, A260188, A267263, A276084, A276086, A276151, A277022, A278226, A283477, A319713, A319715 (inverse Möbius transform), A321683, A324342, A324382, A324383, A324386, A324387, A371091, A373605, A373606, A373607.
Cf. A333426 [k such that a(k)|k], A339215 [numbers not of the form x+a(x) for any x], A358977 [k such that gcd(k, a(k)) = 1].
Cf. A338835, A366693.
Cf. A014601, A042963 (positions of even and odd terms), A343048 (positions of records).
Differs from analogous A034968 for the first time at n=24.

nn = 120; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[# + 1] <= nn &]]; Table[Total@ IntegerDigits[n, b], {n, 0, nn}] (* Version 10.2, or *)
nn = 120; f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Total@ f@ n, {n, 0, 120}] (* Michael De Vlieger, Aug 26 2016 *)

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); }; \\ Antti Karttunen, Feb 27 2019

from sympy import prime, primefactors
def Omega(n): return 0 if n==1 else Omega(n//primefactors(n)[0]) + 1
def a276086(n):
    i=0
    m=pr=1
    while n>0:
        i+=1
        N=prime(i)*pr
        if n%N!=0:
            m*=(prime(i)**((n%N)/pr))
            n-=n%N
        pr=N
    return m
def a(n): return Omega(a276086(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 23 2017

a(n) = 1 + a(A276151(n)) = 1 + a(n-A002110(A276084(n))), a(0) = 0.
or for n >= 1: a(n) = 1 + a(n-A260188(n)).
Other identities and observations. For all n >= 0:
a(n) = A001222(A276086(n)) = A001222(A278226(n)).
a(n) >= A371091(n) >= A267263(n).
From Antti Karttunen, Feb 27 2019: (Start)
a(n) = A000120(A277022(n)).
a(A283477(n)) = A324342(n).
(End)
a(n) = A373606(n) + A373607(n). - Antti Karttunen, Jun 19 2024

A108731 Triangle read by rows: row n gives digits of n in factorial base.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 3, 0, 0, 3, 0, 1, 3, 1, 0, 3, 1, 1, 3, 2, 0, 3, 2, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1

Offset: 0

Franklin T. Adams-Watters, Jun 22 2005

Row lengths are A084558. This sequence contains every finite sequence of nonnegative integers.

T(n,k)=A235168(n,k) for k=0..23; a(n) = A235168(n) for n=0..63, when both tables are seen as flattened lists. - Reinhard Zumkeller, Jan 05 2014

			Triangle begins:
  0
  1
  1, 0
  1, 1
  2, 0
  2, 1
  1, 0, 0
For example, 11 in factorial base is 121 (1*6 + 2*2 + 1*1), so row 11 is 1,2,1.

Reinhard Zumkeller, Rows n = 0..2000 of triangle, flattened
C. A. Laisant, Sur la numération factorielle, application aux permutations, Bulletin de la Société Mathématique de France, 16 (1888), p. 176-183.
Wikipedia, Factorial number system
Index entries for sequences related to factorial base representation

Cf: A000142, A007623, A084558, A301872.

a108731 n k = a108731_row n !! k
a108731_row 0 = [0]
a108731_row n = t n $ reverse $ takeWhile (<= n) $ tail a000142_list
   where t 0 []     = []
         t x (b:bs) = x' : t m bs where (x',m) = divMod x b
a108731_tabf = map a108731_row [0..]
-- Reinhard Zumkeller, Jun 04 2012

b:= proc(n, i) local r; `if`(n b(n, 2)[]:
seq(T(n), n=0..50);  # Alois P. Heinz, Mar 19 2014

b[n_, i_] := b[n, i] = Module[{q, r}, If[n < i, {n}, {q, r} = QuotientRemainder[n, i]; Append[b[q, i + 1], r]]];
T[n_] := b[n, 2];
Table[T[n], {n, 0, 40}] // Flatten (* Jean-François Alcover, Feb 03 2018, after Alois P. Heinz *)

A108731_row(n)={n=[n];while(n[1],n=concat(divrem(n[1],1+#n),n[^1]);n[1]||break);n[^1]~} \\ M. F. Hasler, Jun 20 2017

def row(n, i=2): return [n] if n < i else [*row(n//i, i=i+1), n%i]
print([d for n in range(35) for d in row(n)]) # Michael S. Branicky, Oct 02 2024

Added a(0)=0 and offset changed accordingly by Reinhard Zumkeller, Jun 04 2012

A267263 Number of nonzero digits in representation of n in primorial base.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 1, 2

Offset: 0

Cade Brown, Jan 12 2016

			a(3) = 2 because 3 written in primorial base is 11 with 2 nonzero digits.

Cf. A001221, A002110, A049345, A235168, A276086.
Cf. A060735 (positions of ones).
A060130 is an analogous sequence for the factorial base, from which this differs for the first time at n=30, where a(30) = 1, while A060130(30) = 2.

a:= proc(n) local m, p, r; m, p, r:= n, 2, 0;
       while m>0 do r:= r+`if`(irem(m, p, 'm')>0, 1, 0);
                    p:= nextprime(p)
       od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 15 2016

Table[Length[IntegerDigits[n, MixedRadix@ Prime@ Reverse@ Range@ PrimePi@ n] /. 0 -> Nothing], {n, 0, 120}] (* Michael De Vlieger, Jan 12 2016, Version 10.2 *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Count[f@ n, d_ /; d > 0], {n, 0, 73}] (* Michael De Vlieger, Aug 29 2016 *)

cnz(n) = my(d = digits(n)); sum(k=1, #d, d[k]!=0);
A049345(n, p=2) = if(nA049345(n\p, nextprime(p+1))*10 + n%p)
a(n) = cnz(A049345(n)); \\ Michel Marcus, Jan 12 2016

a(n)=my(s); forprime(p=2,, if(n%p, s++, if(n==0, return(s))); n\=p) \\ Charles R Greathouse IV, Nov 17 2016

a(n) = A001221(A276086(n)). - Antti Karttunen, Aug 21 2016

A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 25, 30, 32, 33, 36, 40, 42, 44, 45, 48, 50, 60, 64, 65, 66, 68, 70, 72, 77, 84, 88, 90, 92, 96, 105, 108, 112, 117, 120, 132, 133, 136, 144, 150, 154, 156, 160, 168, 180, 182, 184, 189, 192, 198, 200, 210, 212, 213, 216, 220

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

Numbers k for which A276086(k) is in A373852. - Antti Karttunen, Jun 22 2024

			1 is a term since A276150(1) = 1 divides 1;
2 is a term since A276150(2) = 1 divides 2;

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A005349, A049345, A049445, A064150, A064438, A064481, A118363, A235168, A276150, A328208, A328212, A373834 (characteristic function)
Cf. A276086, A333427, A333428, A341433, A347496, A358977, A373833, A373852.
Positions of 0's in A373832.

max = 5; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; sumdig[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Select[Range[nmax], Divisible[#, sumdig[#]] &]

isA333426 = A373834; \\ Antti Karttunen, Jun 22 2024

A235224 a(0) = 0, and for n > 0, a(n) = largest k such that A002110(k-1) <= n, where A002110(k) gives the k-th primorial number.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Reinhard Zumkeller, Jan 05 2014

nonn

For n > 0: a(n) = (length of row n in A235168) = A055642(A049345(n)).

For n > 0, a(n) gives the length of primorial base expansion of n. Also, after zero, each value n occurs A061720(n-1) times. - Antti Karttunen, Oct 19 2019

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to primorial base

Cf. A000040, A002110, A049345, A055642, A061395, A061720, A084558, A267263, A276086, A235168, A328114, A328404, A328405, A328406.

a235224 n = length $ takeWhile (<= n) a002110_list

A235224 := proc(n)
    local k;
    if n = 0 then
        0;
    else
        for k from 0 do
            if A002110(k-1) > n then
                return k-1 ;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 19 2021

primorial[n_] := Times @@ Prime[Range[n]];
a[n_] := TakeWhile[primorial /@ Range[0, n], # <= n &] // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 27 2021 *)

A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); }; \\ Antti Karttunen, Oct 19 2019

A235224(n, p=2) = if(!n,n,if(nA235224(n\p, nextprime(p+1)))); \\ (Recursive implementation) - Antti Karttunen, Oct 19 2019

From Antti Karttunen, Oct 19 2019: (Start)
a(n) = A061395(A276086(n)).
For all n >= 0, a(n) >= A267263(n).
For all n >= 1, A000040(a(n)) > A328114(n). (End)

Name corrected to match the data by Antti Karttunen, Oct 19 2019

A343044 Array T(n, k), n, k >= 0, read by antidiagonals; lunar addition table for the primorial base.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 3, 3, 3, 4, 3, 2, 3, 4, 5, 5, 3, 3, 5, 5, 6, 5, 4, 3, 4, 5, 6, 7, 7, 5, 5, 5, 5, 7, 7, 8, 7, 8, 5, 4, 5, 8, 7, 8, 9, 9, 9, 9, 5, 5, 9, 9, 9, 9, 10, 9, 8, 9, 10, 5, 10, 9, 8, 9, 10, 11, 11, 9, 9, 11, 11, 11, 11, 9, 9, 11, 11, 12, 11, 10, 9, 10, 11, 6, 11, 10, 9, 10, 11, 12

Offset: 0

Rémy Sigrist, Apr 05 2021

The i-th digit of T(n, k) in primorial base is the largest of the i-th digits of n and of k in primorial base.

For n = 0..23, the factorial and primorial base representations of n are the same; hence the date sections for this sequence and for A343040 are the same.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12
  ---+----------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12
    1|   1   1   3   3   5   5   7   7   9   9  11  11  13
    2|   2   3   2   3   4   5   8   9   8   9  10  11  14
    3|   3   3   3   3   5   5   9   9   9   9  11  11  15
    4|   4   5   4   5   4   5  10  11  10  11  10  11  16
    5|   5   5   5   5   5   5  11  11  11  11  11  11  17
    6|   6   7   8   9  10  11   6   7   8   9  10  11  12
    7|   7   7   9   9  11  11   7   7   9   9  11  11  13
    8|   8   9   8   9  10  11   8   9   8   9  10  11  14
    9|   9   9   9   9  11  11   9   9   9   9  11  11  15
   10|  10  11  10  11  10  11  10  11  10  11  10  11  16
   11|  11  11  11  11  11  11  11  11  11  11  11  11  17
   12|  12  13  14  15  16  17  12  13  14  15  16  17  12

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the array for n, k < 2*3*5*7*11 (where the color is function of T(n, k))
Rémy Sigrist, PARI program for A343044
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for sequences related to primorial base

Cf. A087061, A235168, A343040, A343045.

PARI
```
See Links section.
```

A333423 Numbers that are palindromes in primorial base.

Original entry on oeis.org

0, 1, 3, 7, 9, 11, 31, 39, 47, 211, 217, 223, 229, 235, 243, 249, 255, 261, 267, 275, 281, 287, 293, 299, 2311, 2347, 2383, 2419, 2455, 2523, 2559, 2595, 2631, 2667, 2735, 2771, 2807, 2843, 2879, 30031, 30061, 30091, 30121, 30151, 30181, 30211, 30247, 30277, 30307

Offset: 1

Amiram Eldar, Mar 20 2020

			3 is a term since its representation in primorial base is 11 (1 * 2# + 1) which is a palindrome.
7 is a term since its representation in primorial base is 101 (1 * 3# + 0 * 2# + 1 = 6 + 1) which is a palindrome.

Amiram Eldar, Table of n, a(n) for n = 1..2500
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A049345, A235168.

Cf. A002113, A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A046807.

max = 6; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Select[Range[0, nmax], PalindromeQ @ IntegerDigits[#, MixedRadix[bases]] &]

A343046 Array T(n, k), n, k >= 0, read by antidiagonals; lunar multiplication table for the primorial base.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 2, 8, 8, 2, 0, 0, 3, 6, 9, 6, 3, 0, 0, 6, 8, 8, 8, 8, 6, 0, 0, 7, 30, 9, 12, 9, 30, 7, 0, 0, 8, 32, 36, 14, 14, 36, 32, 8, 0, 0, 9, 36, 39, 30, 15, 30, 39, 36, 9, 0, 0, 8, 38, 38, 32, 36, 36, 32, 38, 38, 8, 0

Offset: 0

Rémy Sigrist, Apr 05 2021

To compute T(n, k):
- write the primorial base representations of n and of k on two lines, right aligned,
- to "multiply" two digits: take the smallest,
- to "add" two digits: take the largest,
- for example, for T(9, 10):
      9   ->   1 1 1
     10   -> x 1 2 0
             -------
               0 0 0
             1 1 1
         + 1 1 1
         -----------
           1 1 1 1 0  ->  248 = T(9, 10)
See A343044 for the corresponding addition table.

			Array T(n, k) begins:
  n\k|  0  1   2   3   4   5    6    7    8    9   10   11   12
  ---+---------------------------------------------------------
    0|  0  0   0   0   0   0    0    0    0    0    0    0    0
    1|  0  1   2   3   2   3    6    7    8    9    8    9    6  ->  A328841
    2|  0  2   6   8   6   8   30   32   36   38   36   38   30
    3|  0  3   8   9   8   9   36   39   38   39   38   39   36
    4|  0  2   6   8  12  14   30   32   36   38   42   44   60
    5|  0  3   8   9  14  15   36   39   38   39   44   45   66
    6|  0  6  30  36  30  36  210  216  240  246  240  246  210
    7|  0  7  32  39  32  39  216  217  248  249  248  249  216
    8|  0  8  36  38  36  38  240  248  246  248  246  248  240
    9|  0  9  38  39  38  39  246  249  248  249  248  249  246
   10|  0  8  36  38  42  44  240  248  246  248  252  254  270
   11|  0  9  38  39  44  45  246  249  248  249  254  255  276
   12|  0  6  30  36  60  66  210  216  240  246  270  276  420

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, PARI program for A343046
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for sequences related to primorial base

Cf. A087062, A235168, A328841, A343042, A343044, A343047.

PARI
```
See Links section.
```

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = 0.
T(n, 1) = A328841(n).
T(n, n) = A343047(n).

A343404 For any number n with representation (d_w, ..., d_1) in primorial base, a(n) is the least number m such that m mod prime(k) = d_k for k = 1..w (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

0, 1, 4, 1, 2, 5, 6, 21, 16, 1, 26, 11, 12, 27, 22, 7, 2, 17, 18, 3, 28, 13, 8, 23, 24, 9, 4, 19, 14, 29, 120, 15, 190, 85, 50, 155, 36, 141, 106, 1, 176, 71, 162, 57, 22, 127, 92, 197, 78, 183, 148, 43, 8, 113, 204, 99, 64, 169, 134, 29, 30, 135, 100, 205

Offset: 0

Rémy Sigrist, Apr 14 2021

Leading zeros in primorial base expansions are ignored.

The Chinese remainder theorem ensures that this sequence is well defined and provides a way to compute it.

			For n = 42 :
- the expansion of 42 in primary base is "1200",
- so a(42) mod 2 = 0 => a(42) = 2*t for some t >= 0,
     a(42) mod 3 = 0 => a(42) = 6*u for some u >= 0,
     a(42) mod 5 = 2 => a(42) = 12 + 30*v for some v >= 0,
     a(42) mod 7 = 1 => a(42) = 162 + 210*w for some w >= 0,
- we choose w=0 so as to minimize the value,
- hence a(42) = 162.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Wikipedia, Chinese remainder theorem
Index entries for sequences related to primorial base

Cf. A002110, A079276, A143293, A235168, A343405 (fixed points).

a(n) = { my (v=Mod(0,1)); forprime (p=2, oo, if (n==0, return (lift(v)), v=chinese(v, Mod(n, p)); n\=p)) }

a(n) = 1 iff n belongs to A143293.
a(n) = n iff n belongs to A343405.
a(n) < A002110(k) for any n < A002110(k) and k >= 0.
a(A002110(k)) = A079276(k+1) * A002110(k) for any k >= 0.

cp's OEIS Frontend

A049345 n written in primorial base.

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A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

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A108731 Triangle read by rows: row n gives digits of n in factorial base.

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A267263 Number of nonzero digits in representation of n in primorial base.

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A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

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A235224 a(0) = 0, and for n > 0, a(n) = largest k such that A002110(k-1) <= n, where A002110(k) gives the k-th primorial number.

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