cp's OEIS Frontend

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

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