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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..2310

Index entries for sequences related to primorial base

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

CROSSREFS

Cf. A000040, A002110, A049345, A067029, A276084, A276086, A276094, A328569.

Sequence in context: A304779 A334933 A049100 * A030612 A327528 A264857

Adjacent sequences:  A276085 A276086 A276087 * A276089 A276090 A276091

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Aug 22 2016

STATUS

approved

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Last modified July 29 09:41 EDT 2021. Contains 346344 sequences. (Running on oeis4.)