A276073 -id:A276073 - cp's OEIS frontend

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

Antti Karttunen, Aug 18 2016

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.

			   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.

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.

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.

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 *)

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

Name changed by Antti Karttunen, Apr 18 2022

A309064 Langton's ant on a snub square tiling: number of black cells after n moves of the ant when starting on a square.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 10, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 15, 16, 17, 18, 17, 16, 17, 18, 17, 16, 15, 16, 17, 18, 17, 16, 17, 18, 19, 20, 21, 20, 21, 22

Offset: 0

Felix Fröhlich, Jul 10 2019

nonn

First differs from A276073 at n = 16.
On a white square, turn 90 degrees right, flip the color of the tile, then move forward one unit.
On a white triangle, turn 60 degrees right, flip the color of the tile, then move forward one unit.
On a black square, turn 90 degrees left, flip the color of the tile, then move forward one unit.
On a black triangle, turn 60 degrees left, flip the color of the tile, then move forward one unit.

			See illustrations in Fröhlich, 2019.

Lars Blomberg, Table of n, a(n) for n = 0..10000
Lars Blomberg, The state for n=104000, when 872 cells are set
Lars Blomberg, Animation illustrating n=1-3000
Lars Blomberg, Animation illustrating the transition from "chaos" to "avenue", n=96300-99608
Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Wikipedia, Langton's ant
Wikipedia, Snub square tiling

Cf. A255938, A269757, A276073, A308590, A308937, A308973, A326167, A326352.

a(n+1025) = a(n) + 25 for n > 96420. Lars Blomberg, Aug 15 2019

A276074 A276076-polynomials evaluated at X=2 over the field GF(2): a(n) = A248663(A276076(n)).

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1

Offset: 0

Antti Karttunen, Aug 18 2016

Antti Karttunen, Table of n, a(n) for n = 0..5039
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A248663, A276076.

Cf. also A276073, A275808

(define (A276074 n) (A248663 (A276076 n)))

a(n) = A248663(A276076(n)).

A325631 Langton's ant on an elongated triangular tiling: number of black cells after n moves of the ant when starting on a square and initially looking towards one of the edges where that square meets one of the neighboring triangles.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 12, 13, 14, 15, 14, 13, 14, 15, 16, 15, 16, 17, 16, 15, 16, 17, 18, 17, 16, 15, 16, 17, 18, 19, 20, 19, 18, 19, 20, 19, 20, 21, 22, 23, 24, 23, 22, 23, 22, 23, 22, 21, 20, 19, 18, 19, 18, 17, 18, 19, 20

Offset: 0

Felix Fröhlich, Sep 07 2019

nonn

First differs from A276073 at n = 22.
On a white square, turn 90 degrees right, flip the color of the tile, then move forward one unit.
On a white triangle, turn 60 degrees right, flip the color of the tile, then move forward one unit.
On a black square, turn 90 degrees left, flip the color of the tile, then move forward one unit.
On a black triangle, turn 60 degrees left, flip the color of the tile, then move forward one unit.

			See illustrations in Fröhlich, 2019.

Jinyuan Wang, Table of n, a(n) for n = 0..2000
Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Wikipedia, Elongated triangular tiling
Wikipedia, Langton's ant

Cf. A255938, A269757, A308590, A308937, A308973, A326167, A326352, A309064, A309166, A309241, A309279, A309293.

a(n) = a(n-51) + 11 for n >= 1159. - Jinyuan Wang, Jul 15 2025

More terms from Jinyuan Wang, Jul 15 2025

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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.

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A309064 Langton's ant on a snub square tiling: number of black cells after n moves of the ant when starting on a square.

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A276074 A276076-polynomials evaluated at X=2 over the field GF(2): a(n) = A248663(A276076(n)).

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A325631 Langton's ant on an elongated triangular tiling: number of black cells after n moves of the ant when starting on a square and initially looking towards one of the edges where that square meets one of the neighboring triangles.

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