A276086 Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is.

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

Offset: 0

Author

Antti Karttunen, Aug 21 2016

Keywords

• nonn
• base
• look

Comments

Prime product form of primorial base expansion of n.

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 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

From Antti Karttunen, Feb 18 2022: (Start)

The conjecture given in A327969 asks whether applying this function together with the arithmetic derivative (A003415) in some combination or another can eventually transform every positive integer into zero.

Another related open question asks whether there are any other numbers than n=6 such that when starting from that n and by iterating with A003415, one eventually reaches a(n). See comments in A351088.

This sequence is used in A351255 to list the terms of A099308 in a different order, by the increasing exponents of the successive primes in their prime factorization. (End)

From Bill McEachen, Oct 15 2022: (Start)

From inspection, the least significant decimal digits of a(n) terms form continuous chains of 30 as follows. For n == i (mod 30), i=0..5, there are 6 ordered elements of these 8 {1,2,3,6,9,8,7,4}. Then for n == i (mod 30), i=6..29, there are 12 repeated pairs = {5,0}.

Moreover, when the individual elements of any of the possible groups of 6 are transformed via (7*digit) (mod 10), the result matches one of the other 7 groupings (not all 7 may be seen). As example, {1,2,3,6,9,8} transforms to {7,4,1,2,3,6}. (End)

The least significant digit of a(n) in base 4 is given by A353486, and in base 6 by A358840. - Antti Karttunen, Oct 25 2022, Feb 17 2024

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..2310
• Antti Karttunen, Program in LODA-assembly
• Antti Karttunen, Program in LODA-assembly [Cached copy]
• Index entries for sequences related to primorial base

Crossrefs

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

Cf. A000040, A001221, A001222, A002110, 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, A324580 [= n*a(n)], A324895 (largest proper divisor of a(n)), A351252, A353486 (reduced modulo 4), A358840 (modulo 6), A353489, A353516.

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.

Cf. A019565 and A054842 for base-2 and base-10 analogs and A276076 for the analogous "factorial base exp-function", from which this differs for the first time at n=24, where a(24)=625 while A276076(24)=7.

Cf. A327969, A351088, A351458 for sequences with conjectures involving this sequence.

Programs

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 *)
a[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen's Sage code *)

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

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

Python

from sympy import nextprime
def a(n):
    m, p = 1, 2
    while n > 0:
        n, r = divmod(n, p)
        m *= p**r
        p = nextprime(p)
    return m
print([a(n) for n in range(74)])  # Peter Luschny, Apr 20 2024

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.

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))))))

Scheme

(definec (A276086 n) (if (zero? n) 1 (* (expt (A053669 n) (A276088 n)) (A276086 (A276093 n))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

Scheme

(definec (A276086 n) (if (zero? n) 1 (* (A053669 n) (A276086 (- n (A002110 (A276084 n))))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

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]

a(2n+1) = 2 * a(2n). - Antti Karttunen, Feb 17 2022

Extensions

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

Name changed again by Antti Karttunen, Feb 05 2022