A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

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

Offset: 1

Author

Paul Boddington, Jul 21 2005

Keywords

• mult
• easy
• nonn

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

Examples

			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).
		

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.
• Index entries for sequences related to primorial numbers.

Crossrefs

Cf. A319626, A329900 (left inverses).

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

Programs

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-François 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 *)

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 range(1, 101)]) # Indranil Ghosh, May 14 2017

Sage

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

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) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]

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)

From Antti Karttunen, Nov 19 2019: (Start)

Further identities:

a(A307035(n)) = A000142(n).

a(A003418(n)) = A181814(n).

a(A025487(n)) = A181817(n).

a(A181820(n)) = A181822(n).

a(A019565(n)) = A283477(n).

A001221(a(n)) = A061395(n).

A001222(a(n)) = A056239(n).

A181819(a(n)) = A122111(n).

A124859(a(n)) = A181821(n).

A085082(a(n)) = A238690(n).

A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)

A000188(a(n)) = A329602(n). (square root of the greatest square divisor)

A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)

A005361(a(n)) = A329382(n). (product of exponents of prime factors)

A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)

A000005(a(n)) = A329605(n). (number of divisors)

A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)

A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)

A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)

A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)

A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)

A324580(a(n)) = A324887(n).

A276150(a(n)) = A324888(n). (digit sum in primorial base)

A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)

A243055(a(n)) = A329343(n).

A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)

A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)

A328114(a(n)) = A329344(n). (maximal digit in primorial base)

A062977(a(n)) = A325226(n).

A097248(a(n)) = A283478(n).

A324895(a(n)) = A324896(n).

A324655(a(n)) = A329046(n).

A327860(a(n)) = A329047(n).

A329601(a(n)) = A329607(n).

(End)

a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019

From Antti Karttunen, Jul 09 2021: (Start)

a(n) = A346092(n) + A346093(n).

a(n) = A346108(n) - A346109(n).

a(A342012(n)) = A004490(n).

a(A337478(n)) = A336389(n).

A336835(a(n)) = A337474(n).

A342002(a(n)) = A342920(n).

A328571(a(n)) = A346091(n).

A328572(a(n)) = A344592(n).

(End)

Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

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

Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020