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A276076 Prime-factorization representations of "factorial base digit polynomials": a(0) = 1, for n >= 1, a(n) = A275733(n) * a(A276009(n)). 20
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 (list; graph; refs; listen; history; text; internal format)
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).

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.

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, A019565, A033312, A034968, A048675, A051903, A059590, A060130, A076954, A246359, A248663, A276073, A276074.

Cf. also A275733, A275734, A275735, A275725 for other such encodings of factorial base related polynomials.

Sequence in context: A018251 A218339 A329248 * A276086 A018402 A018441

Adjacent sequences:  A276073 A276074 A276075 * A276077 A276078 A276079

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Aug 18 2016

STATUS

approved

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Last modified December 5 23:39 EST 2019. Contains 329784 sequences. (Running on oeis4.)