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A276086 Digits in primorial base representation of n become the exponents of successive primes that are multiplied together: a(0)=1, a(n) = A053669(n)*a(A276151(n)). 198
1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 625, 1250, 1875, 3750, 5625, 11250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 4375, 8750, 13125, 26250, 39375, 78750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Sequence contains only terms of A048103 and each term of A048103 occurs exactly once.

LINKS

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

Index entries for sequences related to primorial base

FORMULA

a(0) = 1; for n >= 1, a(n) = A053669(n) * a(n-A002110(A276084(n))).

a(0) = 1; for n >= 1, a(n) = A053669(n)^A276088(n) * a(A276093(n)).

Other identities. For all n >= 0:

A276085(a(n)) = n.

A001221(a(n)) = A267263(n).

A001222(a(n)) = A276150(n).

A071178(a(n)) = A276153(n).

a(A002110(n)) = A000040(n+1).

a(A143293(n)) = A002110(n+1).

a(A057588(n)) = A276092(n).

a(A276156(n)) = A019565(n).

A046523(a(n)) = A278226(n).

For all n >= 1:

A020639(a(n)) = A053669(n). [The smallest prime not dividing n is mapped to the smallest prime dividing n.]

A055396(a(n)) = A257993(n).

EXAMPLE

For n = 24, which has primorial base representation (see A049345) "400" as 24 = 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*6 + 0*2 + 0*1, thus a(24) = prime(3)^4 * prime(2)^0 * prime(1)^0 = 5^4 = 625.

For n = 35 = "1021" as 35 = 1*A002110(3) + 0*A002110(2) + 2*A002110(1) + 1*A002110(0) = 1*30 + 0*6 + 2*2 + 1*1, thus a(35) = prime(4)^1 * prime(2)^2 * prime(1) = 7*3*3*2 = 126.

MATHEMATICA

b = MixedRadix[Reverse@ Prime@ Range@ 12]; Table[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[n, b], {n, 0, 51}] (* Michael De Vlieger, Aug 23 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[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ f@ n], {n, 0, 73}] (* Michael De Vlieger, Aug 30 2016, Pre-Version 10 *)

PROG

(PARI) A276086(n) = { my(i=0, m=1, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m*=(prime(i)^((n%nextpr)/pr)); n-=(n%nextpr)); pr=nextpr); m; }; \\ Antti Karttunen, May 12 2017

(Scheme)

(define (A276086 n) (let loop ((n n) (t 1) (i 1)) (if (zero? n) t (let* ((p (A000040 i)) (d (modulo n p))) (loop (/ (- n d) p) (* t (expt p d)) (+ 1 i))))))

;; A version following the given recurrence:

(definec (A276086 n) (if (zero? n) 1 (* (expt (A053669 n) (A276088 n)) (A276086 (A276093 n)))))

;; Or even simpler:

(definec (A276086 n) (if (zero? n) 1 (* (A053669 n) (A276086 (- n (A002110 (A276084 n)))))))

(Python)

from sympy import prime

def a(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 # Indranil Ghosh, May 12 2017, after Antti Karttunen's PARI code

CROSSREFS

Cf. A000040, A001221, A001222, A002110, A019565, A020639, A049345, A053669, A055396, A057588, A071178, A143293, A257993, A267263, A276084, A276088, A276092, A276093, A276147, A276150, A276151, A276153, A276156.

Cf. A276085 (a left inverse) and also A276087.

Cf. A048103 (terms sorted into ascending order), A100716 (natural numbers not present in this sequence).

Differs from related A276076 for the first time at n=24, where a(24)=625 while A276076(24)=7.

Cf. A054842 for base-10 analog.

Cf. A278226 (associated filter-sequence), A286626 (and its rgs-version).

Sequence in context: A218339 A329248 A276076 * A018402 A018441 A124879

Adjacent sequences:  A276083 A276084 A276085 * A276087 A276088 A276089

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Aug 21 2016

STATUS

approved

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Last modified November 12 09:29 EST 2019. Contains 329054 sequences. (Running on oeis4.)