|
%I #115 Aug 04 2022 05:54:46
%S 1,2,6,4,30,12,210,8,36,60,2310,24,30030,420,180,16,510510,72,9699690,
%T 120,1260,4620,223092870,48,900,60060,216,840,6469693230,360,
%U 200560490130,32,13860,1021020,6300,144,7420738134810,19399380,180180,240,304250263527210,2520
%N Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).
%C This sequence is a permutation of A025487.
%C And thus also a permutation of A181812, see the formula section. - _Antti Karttunen_, Jul 21 2014
%C 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
%H Amiram Eldar, <a href="/A108951/b108951.txt">Table of n, a(n) for n = 1..2370</a> (terms 1..256 from Antti Karttunen)
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>.
%F Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
%F 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]
%F From _Antti Karttunen_, Jul 21 2014: (Start)
%F a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
%F a(n) = n * A181811(n).
%F a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
%F a(n) = A181812(A048673(n)).
%F Other identities:
%F A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
%F A071178(a(n)) = A071178(n). [And also its exponent.]
%F a(2^n) = 2^n. [Fixes the powers of two.]
%F 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.]
%F (End)
%F From _Antti Karttunen_, Nov 19 2019: (Start)
%F Further identities:
%F a(A307035(n)) = A000142(n).
%F a(A003418(n)) = A181814(n).
%F a(A025487(n)) = A181817(n).
%F a(A181820(n)) = A181822(n).
%F a(A019565(n)) = A283477(n).
%F A001221(a(n)) = A061395(n).
%F A001222(a(n)) = A056239(n).
%F A181819(a(n)) = A122111(n).
%F A124859(a(n)) = A181821(n).
%F A085082(a(n)) = A238690(n).
%F A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
%F A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
%F A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
%F A005361(a(n)) = A329382(n). (product of exponents of prime factors)
%F A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
%F A000005(a(n)) = A329605(n). (number of divisors)
%F A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
%F A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
%F A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
%F A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
%F A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
%F A324580(a(n)) = A324887(n).
%F A276150(a(n)) = A324888(n). (digit sum in primorial base)
%F A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
%F A243055(a(n)) = A329343(n).
%F A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
%F A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
%F A328114(a(n)) = A329344(n). (maximal digit in primorial base)
%F A062977(a(n)) = A325226(n).
%F A097248(a(n)) = A283478(n).
%F A324895(a(n)) = A324896(n).
%F A324655(a(n)) = A329046(n).
%F A327860(a(n)) = A329047(n).
%F A329601(a(n)) = A329607(n).
%F (End)
%F a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - _Antti Karttunen_, Dec 29 2019
%F From _Antti Karttunen_, Jul 09 2021: (Start)
%F a(n) = A346092(n) + A346093(n).
%F a(n) = A346108(n) - A346109(n).
%F a(A342012(n)) = A004490(n).
%F a(A337478(n)) = A336389(n).
%F A336835(a(n)) = A337474(n).
%F A342002(a(n)) = A342920(n).
%F A328571(a(n)) = A346091(n).
%F A328572(a(n)) = A344592(n).
%F (End)
%F Sum_{n>=1} 1/a(n) = A161360. - _Amiram Eldar_, Aug 04 2022
%e a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
%e a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).
%t 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 *)
%t 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 *)
%o (Scheme, with Antti Karttunen's IntSeq-library for memoizing definec-macro)
%o (definec (A108951 n) (if (= 1 n) n (* n (A108951 (A064989 n)))))
%o ;; _Antti Karttunen_, Jul 21 2014
%o (Sage)
%o def sharp_primorial(n): return sloane.A002110(prime_pi(n))
%o def p(f):
%o return sharp_primorial(f[0])^f[1]
%o [prod(p(f) for f in factor(n)) for n in range (1,51)]
%o # _Giuseppe Coppoletta_, Feb 07 2015
%o (PARI) primorial(n)=prod(i=1,primepi(n),prime(i))
%o a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ _Charles R Greathouse IV_, Jun 28 2015
%o (Python)
%o from sympy import primerange, factorint
%o from operator import mul
%o def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
%o def a(n):
%o f = factorint(n)
%o return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, May 14 2017
%Y Cf. A319626, A329900 (left inverses).
%Y 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.
%K mult,easy,nonn
%O 1,2
%A _Paul Boddington_, Jul 21 2005
%E More terms computed by _Antti Karttunen_, Jul 21 2014
%E 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
%E Name "Primorial inflation" (coined by _Matthew Vandermast_ in A181815) prefixed to the name by _Antti Karttunen_, Jan 14 2020
| 