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A276088
The least significant nonzero digit in primorial base representation of n: a(n) = A276094(n) / A002110(A276084(n)) (with a(0) = 0).
12
0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 4
OFFSET
0,5
COMMENTS
For any n >= 1, start from k = n and repeatedly try to divide as many successive primes as possible out of k, iterating as k/2 -> k, k/3 -> k, k/5 -> k, until a nonzero remainder is encountered, which is then the value of a(n). (See the last example).
Note that the sequence has been defined so that it will eventually include also "digits" (actually: value holders) > 9 that occur as the least significant nonzero digits in primorial base representation. Thus any eventual decimal corruption of A049345 will not affect these values.
The sums of the first 10^k terms (starting from n=1), for k = 1, 2, ..., are 12, 138, 1441, 14565, 145950, 1459992, 14600211, 146002438, 1460025336, 14600254674, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.460025... . - Amiram Eldar, Sep 10 2022
FORMULA
a(0) = 0, and for n >= 1, a(n) = A276094(n) / A002110(A276084(n)).
From Antti Karttunen, Oct 29 2019: (Start)
a(n) = A067029(A276086(n)).
a(A276086(n)) = A328569(n).
(End).
EXAMPLE
n A049345 the rightmost nonzero = a(n)
---------------------------------------------------------
0 0 0
1 1 1
2 10 1
3 11 1
4 20 2
5 21 1
6 100 1
7 101 1
8 110 1
9 111 1
10 120 2
11 121 1
12 200 2
13 201 1
14 210 1
15 211 1
16 220 2
.
For n=48 according to the iteration interpretation, we obtain first 48/2 = 24, and the remainder is zero, so we continue: 24/3 = 8 and here the remainder is zero as well, so we try next 8/5, but this gives the nonzero remainder 3, thus a(48)=3.
For n=2100, which could be written "A0000" in primorial base (where A stands for digit "ten", as 2100 = 10*A002110(4)), the least significant nonzero value holder (also the most significant) is thus 10 and a(2100) = 10. (The first point where this sequence attains a value larger than 9).
MATHEMATICA
nn = 120; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[Last[IntegerDigits[n, b] /. 0 -> Nothing, 0], {n, 0, nn}] (* Version 11, or *)
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]]]; {0}~Join~Table[Last@ DeleteCases[f@ n, d_ /; d == 0], {n, 120}] (* Michael De Vlieger, Aug 30 2016 *)
PROG
(PARI) A276088(n) = { my(e=0, p=2); while(n && !(e=(n%p)), n = n/p; p = nextprime(1+p)); (e); }; \\ Antti Karttunen, Oct 29 2019
(Scheme, two versions)
(define (A276088 n) (if (zero? n) n (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) d (loop (/ (- n d) p) (+ 1 i)))))))
(define (A276088 n) (if (zero? n) n (/ (A276094 n) (A002110 (A276084 n)))))
(Python)
from sympy import nextprime, primepi, primorial
def a053669(n):
p = 2
while True:
if n%p!=0: return p
else: p=nextprime(p)
def a257993(n): return primepi(a053669(n))
def a002110(n): return 1 if n<1 else primorial(n)
def a276094(n): return 0 if n==0 else n%a002110(a257993(n))
def a(n): return 0 if n==0 else a276094(n)//a002110(a257993(n) - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Aug 22 2016
STATUS
approved