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

1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 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

Offset

0,2

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 0..5040

Indranil Ghosh, Python program for computing this sequence.

Index entries for sequences related to factorial base representation.

Formula

a(0) = 1, for n >= 1, a(n) = A275733(n) * a(A276009(n)).

Or: for n >= 1, a(n) = a(A257687(n)) * A000040(A084558(n))^A099563(n).

Other identities.

For all n >= 0:

A276075(a(n)) = n.

A001221(a(n)) = A060130(n).

A001222(a(n)) = A034968(n).

A051903(a(n)) = A246359(n).

A048675(a(n)) = A276073(n).

A248663(a(n)) = A276074(n).

a(A007489(n)) = A002110(n).

a(A059590(n)) = A019565(n).

For all n >= 1:

a(A000142(n)) = A000040(n).

a(A033312(n)) = A076954(n-1).

From Antti Karttunen, Apr 18 2022: (Start)

a(n) = A276086(A351576(n)).

A276085(a(n)) = A351576(n)

A003557(a(n)) = A351577(n).

A003415(a(n)) = A351950(n).

A069359(a(n)) = A351951(n).

A083345(a(n)) = A342001(a(n)) = A351952(n).

A351945(a(n)) = A351954(n).

A181819(a(n)) = A275735(n).

(End)

lambda(a(n)) = A262725(n+1), where lambda is Liouville's function, A008836. - Antti Karttunen and Peter Munn, Aug 09 2024

Example

   n  A007623   polynomial     encoded as             a(n)
   -------------------------------------------------------
   0       0    0-polynomial   (empty product)        = 1
   1       1    1*x^0          prime(1)^1             = 2
   2      10    1*x^1          prime(2)^1             = 3
   3      11    1*x^1 + 1*x^0  prime(2) * prime(1)    = 6
   4      20    2*x^1          prime(2)^2             = 9
   5      21    2*x^1 + 1*x^0  prime(2)^2 * prime(1)  = 18
   6     100    1*x^2          prime(3)^1             = 5
   7     101    1*x^2 + 1*x^0  prime(3) * prime(1)    = 10
and:
  23     321  3*x^2 + 2*x + 1  prime(3)^3 * prime(2)^2 * prime(1)
                                      = 5^3 * 3^2 * 2 = 2250.

Mathematica

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 *)

Prog

(Scheme, two versions)
(define (A276076 n) (if (zero? n) 1 (* (expt (A000040 (A084558 n)) (A099563 n)) (A276076 (A257687 n)))))
(define (A276076 n) (if (zero? n) 1 (* (A275733 n) (A276076 (A276009 n)))))

Crossrefs

Cf. A000040, A007623, A084558, A099563, A257687, A276009.

Cf. A276075 (a left inverse).

Cf. A276078 (same terms in ascending order).

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.

Cf. also A275733, A275734, A275735, A275725 for other such encodings of factorial base related polynomials, and A276086 for a primorial base analog.

Sequence in context: A018251 A218339 A329248 * A276086 A346101 A351255

Adjacent sequences: A276073 A276074 A276075 * A276077 A276078 A276079

Keyword

nonn,base

Author

Antti Karttunen, Aug 18 2016

Extensions

Name changed by Antti Karttunen, Apr 18 2022

Status

approved