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A276086 Prime product form of primorial base expansion of n: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is. 221
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 is a permutation of A048103. It maps the smallest prime not dividing n to the smallest prime dividing n, that is, A020639(a(n)) = A053669(n) holds for all n >= 1.

The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever A329041(x,y) = 1, that is, when the adding x and y together will not generate any carries in the primorial base. Examples of such pairs of x and y are A328841(n) & A328842(n), and also A328770(n) (when added with itself). - Antti Karttunen, Oct 31 2019

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(A276151(n)) = A053669(n) * a(n-A002110(A276084(n))).

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

a(n) = A328841(a(n)) + A328842(a(n)) = A328843(n) + A328844(n).

a(n) = a(A328841(n)) * a(A328842(n)) = A328571(n) * A328572(n).

a(n) = A328475(n) * A328580(n) = A328476(n) + A328580(n).

a(A002110(n)) = A000040(n+1). [Maps primorials to primes]

a(A143293(n)) = A002110(n+1). [Maps partial sums of primorials to primorials]

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

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

a(A283477(n)) = A324289(n).

a(A003415(n)) = A327859(n).

Here the text in brackets shows how the right hand side sequence is a function of the primorial base expansion of n:

A001221(a(n)) = A267263(n). [Number of nonzero digits]

A001222(a(n)) = A276150(n). [Sum of digits]

A067029(a(n)) = A276088(n). [The least significant nonzero digit]

A071178(a(n)) = A276153(n). [The most significant digit]

A061395(a(n)) = A235224(n). [Number of significant digits]

A051903(a(n)) = A328114(n). [Largest digit]

A055396(a(n)) = A257993(n). [Number of trailing zeros + 1]

A257993(a(n)) = A328570(n). [Index of the least significant zero digit]

A079067(a(n)) = A328620(n). [Number of nonleading zeros]

A056169(a(n)) = A328614(n). [Number of 1-digits]

A056170(a(n)) = A328615(n). [Number of digits larger than 1]

A277885(a(n)) = A328828(n). [Index of the least significant digit > 1]

A134193(a(n)) = A329028(n). [The least missing nonzero digit]

A005361(a(n)) = A328581(n). [Product of nonzero digits]

A072411(a(n)) = A328582(n). [LCM of nonzero digits]

A001055(a(n)) = A317836(n). [Number of carry-free partitions of n in primorial base]

Various number theoretical functions applied:

A000005(a(n)) = A324655(n). [Number of divisors of a(n)]

A000203(a(n)) = A324653(n). [Sum of divisors of a(n)]

A000010(a(n)) = A324650(n). [Euler phi applied to a(n)]

A023900(a(n)) = A328583(n). [Dirichlet inverse of Euler phi applied to a(n)]

A069359(a(n)) = A329029(n). [Sum a(n)/p over primes p dividing a(n)]

A003415(a(n)) = A327860(n). [Arithmetic derivative of a(n)]

Other identities:

A276085(a(n)) = n.          [A276085 is a left inverse]

A020639(a(n)) = A053669(n). [The smallest prime not dividing n -> the smallest prime dividing n]

A046523(a(n)) = A278226(n). [Least number with the same prime signature as a(n)]

A246277(a(n)) = A329038(n).

A181819(a(n)) = A328835(n).

A053669(a(n)) = A326810(n), A326810(a(n)) = A328579(n).

A257993(a(n)) = A328570(n), A328570(a(n)) = A328578(n).

A328613(a(n)) = A328763(n), A328620(a(n)) = A328766(n).

A328828(a(n)) = A328829(n).

A053589(a(n)) = A328580(n). [Greatest primorial number which divides a(n)]

A276151(a(n)) = A328476(n). [... and that primorial subtracted from a(n)]

A111701(a(n)) = A328475(n).

A328114(a(n)) = A328389(n). [Greatest digit of primorial base expansion of a(n)]

A328389(a(n)) = A328394(n), A328394(a(n)) = A328398(n).

A235224(a(n)) = A328404(n), A328405(a(n)) = A328406(n).

a(A328625(n)) = A328624(n), a(A328626(n)) = A328627(n). ["Twisted" variants]

a(A108951(n)) = A324886(n).

a(n) mod n = A328386(n).

a(a(n)) = A276087(n), a(a(a(n))) = A328403(n). [2- and 3-fold applications]

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

(PARI) A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); }; \\ (Better than above one, avoids unnecessary construction of primorials). - Antti Karttunen, Oct 14 2019

(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

(Sage)

def A276086(n):

    m=1

    i=1

    while n>0:

        p = sloane.A000040(i)

        m *= (p**(n%p))

        n = floor(n/p)

        i += 1

    return (m)

# Antti Karttunen, Oct 14 2019, after Indranil Ghosh's Python code above, and my own leaner PARI code from Oct 14 2019. This avoids unnecessary construction of primorials.

CROSSREFS

Cf. A276085 (a left inverse) and also A276087, A328403.

Cf. A000040, A001221, A001222, A002110, A019565, A020639, A049345, A053669, A055396, A057588, A071178, A143293, A257993, A267263, A276084, A276088, A276092, A276093, A276147, A276150, A276151, A276153, A276156, A283477, A324198 (= gcd(n, a(n))), A328584 (= lcm(n, a(n))), A324646, A324289, A328386, A328403, A328475, A328571, A328572, A328578, A328612, A328613, A328620, A328624, A328627, A328763, A328766, A328828, A328835, A328841, A328842, A328843, A328844, A329041.

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

Cf. A278226 (associated filter-sequence), A286626 (and its rgs-version), A328477.

Cf. A328316 (iterates started from zero).

Cf. A327858, A327859, A327860, A327963, A328097, A328098, A328099, A328110, A328112, A328382 for various combinations with arithmetic derivative (A003415).

Cf. also A327167, A329037.

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.

Sequence in context: A218339 A329248 A276076 * A018402 A018441 A124879

Adjacent sequences:  A276083 A276084 A276085 * A276087 A276088 A276089

KEYWORD

nonn,base,look

AUTHOR

Antti Karttunen, Aug 21 2016

EXTENSIONS

Name edited and new link-formulas added by Antti Karttunen, Oct 29 2019

STATUS

approved

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Last modified December 1 00:44 EST 2020. Contains 338831 sequences. (Running on oeis4.)