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

Author

Antti Karttunen, Aug 18 2016

Keywords

• nonn
• base

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.

Examples

			   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.
		

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.

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.

Programs

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 *)

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

Extensions

Name changed by Antti Karttunen, Apr 18 2022