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A108951 Completely multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x). 31
1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence is a permutation of A025487.

And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014

A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..2370 (terms 1..256 from Antti Karttunen)

Index to divisibility sequences

Index entries for sequences computed from indices in prime factorization

FORMULA

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...

Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]

From Antti Karttunen, Jul 21 2014: (Start)

a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).

a(n) = n * A181811(n).

a(n) = A181812(A048673(n)).

Other identities:

A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]

A071178(a(n)) = A071178(n). [And also its exponent.]

a(2^n) = 2^n. [Fixes the powers of two.]

A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]

(End)

EXAMPLE

a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24

a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

MATHEMATICA

a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-Fran├žois Alcover, Feb 24 2015 *)

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)

PROG

(Scheme, with Antti Karttunen's IntSeq-library for memoizing definec-macro)

(definec (A108951 n) (if (= 1 n) n (* n (A108951 (A064989 n)))))

;; Antti Karttunen, Jul 21 2014

(Sage) def sharp_primorial(n): return sloane.A002110(prime_pi(n))

def p(n, k): return sharp_primorial(factor(n)[k][0])^factor(n)[k][1];

[prod(p(n, k) for k in range (len(factor(n)))) for n in range (1, 51)] # Giuseppe Coppoletta, Feb 07 2015

(PARI) primorial(n)=prod(i=1, primepi(n), prime(i))

a(n)=my(f=factor(n)); prod(i=1, #f~, primorial(f[i, 1])^f[i, 2]) \\ Charles R Greathouse IV, Jun 28 2015

(Python)

from sympy import primerange, factorint

from operator import mul

def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])

def a(n):

    f = factorint(n)

    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])

print [a(n) for n in xrange(1, 101)] # Indranil Ghosh, May 14 2017

CROSSREFS

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A124859, A181811, A181812, A283477, A283478.

Sequence in context: A324922 A064538 A002790 * A181822 A174940 A293011

Adjacent sequences:  A108948 A108949 A108950 * A108952 A108953 A108954

KEYWORD

mult,easy,nonn

AUTHOR

Paul Boddington, Jul 21 2005

EXTENSIONS

More terms computed by Antti Karttunen, Jul 21 2014

The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015

STATUS

approved

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Last modified October 14 16:48 EDT 2019. Contains 328022 sequences. (Running on oeis4.)