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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.

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

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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).