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A276076 Factorial base exp-function: digits in factorial base representation of n become the exponents of successive prime factors whose product a(n) is.

%I A276076 #40 Aug 11 2024 10:34:56
%S A276076 1,2,3,6,9,18,5,10,15,30,45,90,25,50,75,150,225,450,125,250,375,750,
%T A276076 1125,2250,7,14,21,42,63,126,35,70,105,210,315,630,175,350,525,1050,
%U A276076 1575,3150,875,1750,2625,5250,7875,15750,49,98,147,294,441,882,245,490,735,1470,2205,4410,1225,2450,3675,7350,11025,22050,6125,12250,18375,36750,55125,110250,343
%N A276076 Factorial base exp-function: digits in factorial base representation of n become the exponents of successive prime factors whose product a(n) is.
%C A276076 These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the digit in one-based position k of the factorial base representation of n. See the examples.
%H A276076 Antti Karttunen, <a href="/A276076/b276076.txt">Table of n, a(n) for n = 0..5040</a>
%H A276076 Indranil Ghosh, <a href="/A276076/a276076_1.txt">Python program for computing this sequence</a>.
%H A276076 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%F A276076 a(0) = 1, for n >= 1, a(n) = A275733(n) * a(A276009(n)).
%F A276076 Or: for n >= 1, a(n) = a(A257687(n)) * A000040(A084558(n))^A099563(n).
%F A276076 Other identities.
%F A276076 For all n >= 0:
%F A276076 A276075(a(n)) = n.
%F A276076 A001221(a(n)) = A060130(n).
%F A276076 A001222(a(n)) = A034968(n).
%F A276076 A051903(a(n)) = A246359(n).
%F A276076 A048675(a(n)) = A276073(n).
%F A276076 A248663(a(n)) = A276074(n).
%F A276076 a(A007489(n)) = A002110(n).
%F A276076 a(A059590(n)) = A019565(n).
%F A276076 For all n >= 1:
%F A276076 a(A000142(n)) = A000040(n).
%F A276076 a(A033312(n)) = A076954(n-1).
%F A276076 From _Antti Karttunen_, Apr 18 2022: (Start)
%F A276076 a(n) = A276086(A351576(n)).
%F A276076 A276085(a(n)) = A351576(n)
%F A276076 A003557(a(n)) = A351577(n).
%F A276076 A003415(a(n)) = A351950(n).
%F A276076 A069359(a(n)) = A351951(n).
%F A276076 A083345(a(n)) = A342001(a(n)) = A351952(n).
%F A276076 A351945(a(n)) = A351954(n).
%F A276076 A181819(a(n)) = A275735(n).
%F A276076 (End)
%F A276076 lambda(a(n)) = A262725(n+1), where lambda is Liouville's function, A008836. - _Antti Karttunen_ and _Peter Munn_, Aug 09 2024
%e A276076    n  A007623   polynomial     encoded as             a(n)
%e A276076    -------------------------------------------------------
%e A276076    0       0    0-polynomial   (empty product)        = 1
%e A276076    1       1    1*x^0          prime(1)^1             = 2
%e A276076    2      10    1*x^1          prime(2)^1             = 3
%e A276076    3      11    1*x^1 + 1*x^0  prime(2) * prime(1)    = 6
%e A276076    4      20    2*x^1          prime(2)^2             = 9
%e A276076    5      21    2*x^1 + 1*x^0  prime(2)^2 * prime(1)  = 18
%e A276076    6     100    1*x^2          prime(3)^1             = 5
%e A276076    7     101    1*x^2 + 1*x^0  prime(3) * prime(1)    = 10
%e A276076 and:
%e A276076   23     321  3*x^2 + 2*x + 1  prime(3)^3 * prime(2)^2 * prime(1)
%e A276076                                       = 5^3 * 3^2 * 2 = 2250.
%t A276076 a[n_] := Module[{k = n, m = 2, r, p = 2, q = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, q *= p^r; p = NextPrime[p]; m++]; q]; Array[a, 100, 0] (* _Amiram Eldar_, Feb 07 2024 *)
%o A276076 (Scheme, two versions)
%o A276076 (define (A276076 n) (if (zero? n) 1 (* (expt (A000040 (A084558 n)) (A099563 n)) (A276076 (A257687 n)))))
%o A276076 (define (A276076 n) (if (zero? n) 1 (* (A275733 n) (A276076 (A276009 n)))))
%Y A276076 Cf. A000040, A007623, A084558, A099563, A257687, A276009.
%Y A276076 Cf. A276075 (a left inverse).
%Y A276076 Cf. A276078 (same terms in ascending order).
%Y A276076 Cf. also A000142, A001221, A001222, A002110, A007489, A008836, A019565, A033312, A034968, A048675, A051903, A059590, A060130, A076954, A246359, A248663, A262725, A276073, A276074, A351576, A351577, A351950, A351951, A351952, A351954.
%Y A276076 Cf. also A275733, A275734, A275735, A275725 for other such encodings of factorial base related polynomials, and A276086 for a primorial base analog.
%K A276076 nonn,base
%O A276076 0,2
%A A276076 _Antti Karttunen_, Aug 18 2016
%E A276076 Name changed by _Antti Karttunen_, Apr 18 2022