cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A275725 a(n) = A275723(A002110(1+A084558(n)), n); prime factorization encodings of cycle-polynomials computed for finite permutations listed in the order that is used in tables A060117 / A060118.

Original entry on oeis.org

2, 4, 18, 8, 12, 8, 150, 100, 54, 16, 24, 16, 90, 40, 54, 16, 36, 16, 60, 40, 36, 16, 24, 16, 1470, 980, 882, 392, 588, 392, 750, 500, 162, 32, 48, 32, 270, 80, 162, 32, 108, 32, 120, 80, 72, 32, 48, 32, 1050, 700, 378, 112, 168, 112, 750, 500, 162, 32, 48, 32, 450, 200, 162, 32, 72, 32, 300, 200, 108, 32, 48, 32, 630, 280, 378, 112, 252, 112, 450, 200
Offset: 0

Views

Author

Antti Karttunen, Aug 09 2016

Keywords

Comments

In this context "cycle-polynomials" are single-variable polynomials where the coefficients (encoded with the exponents of prime factorization of n) are equal to the lengths of cycles in the permutation listed with index n in tables A060117 or A060118. See the examples.

Examples

			Consider the first eight permutations (indices 0-7) listed in A060117:
  1 [Only the first 1-cycle explicitly listed thus a(0) = 2^1 = 2]
  2,1 [One transposition (2-cycle) in beginning, thus a(1) = 2^2 = 4]
  1,3,2 [One fixed element in beginning, then transposition, thus a(2) = 2^1 * 3^2 = 18]
  3,1,2 [One 3-cycle, thus a(3) = 2^3 = 8]
  3,2,1 [One transposition jumping over a fixed element, a(4) = 2^2 * 3^1 = 12]
  2,3,1 [One 3-cycle, thus a(5) = 2^3 = 8]
  1,2,4,3 [Two 1-cycles, then a 2-cycle, thus a(6) = 2^1 * 3^1 * 5^2 = 150].
  2,1,4,3 [Two 2-cycles, not crossed, thus a(7) = 2^2 * 5^2 = 100]
and also the seventeenth one at n=16 [A007623(16)=220] where we have:
  3,4,1,2 [Two 2-cycles crossed, thus a(16) = 2^2 * 3^2 = 36].

Links

Crossrefs

Cf. A000040, A001222, A001221, A002110, A007814, A046660, A048675, A051903, A056169, A056170, A084558, A243054, A257510, A275723, A275803, A275832.
Cf. A275807 (terms divided by 2).
Cf. also A060129, A060128, A060130, A060131, A072411, A275726, A275812, A275851.
Cf. also A275733, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

Programs

Formula

a(n) = A275723(A002110(1+A084558(n)), n).
Other identities:
A001221(a(n)) = 1+A257510(n) (for all n >= 1).
A001222(a(n)) = 1+A084558(n).
A007814(a(n)) = A275832(n).
A048675(a(n)) = A275726(n).
A051903(a(n)) = A275803(n).
A056169(a(n)) = A275851(n).
A046660(a(n)) = A060130(n).
A072411(a(n)) = A060131(n).
A056170(a(n)) = A060128(n).
A275812(a(n)) = A060129(n).
a(n!) = 2 * A243054(n) = A000040(n)*A002110(n) for all n >= 1.

A275724 Transpose of square array A275723.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 09 2016

Keywords

Comments

See A275723.

Links

Crossrefs

Transpose: A275723.
Cf. A002260, A004736, A038722.

Programs

Formula

a(n) = A275723(A038722(n)). [When A275723 and A275724 are considered as one-dimensional sequences.]

A273673 Square array A(n,k) = (n / prime(1+A084558(k))^e) * prime(1+A084558(k)-A099563(k))^e, where e = A249344((1+A084558(k)), n) = the exponent of the largest power of prime(1+A084558(k)) which divides n. Array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 3, 4, 1, 2, 3, 4, 3, 6, 7, 1, 2, 3, 4, 2, 6, 7, 8, 1, 2, 3, 4, 2, 6, 7, 8, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 6, 11, 1, 2, 3, 4, 5, 6, 5, 8, 9, 6, 11, 8, 1, 2, 3, 4, 5, 6, 5, 8, 9, 4, 11, 12, 13, 1, 2, 3, 4, 5, 6, 5, 8, 9, 4, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 5, 8, 9, 10, 11, 12, 13, 14, 10
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2016

Keywords

Comments

Informally: "clear" the exponent of prime(1+A084558(k)) and add it (the old value of exponent) to the exponent of prime(1+A084558(k)-A099563(k)) in the prime factorization of n.
Auxiliary function for computing array A275723.

Examples

			The top left 6 x 15 corner of the array:
   1,  1,  1,  1,  1,  1
   2,  2,  2,  2,  2,  2
   2,  3,  3,  3,  3,  3
   4,  4,  4,  4,  4,  4
   5,  3,  3,  2,  2,  5
   4,  6,  6,  6,  6,  6
   7,  7,  7,  7,  7,  5
   8,  8,  8,  8,  8,  8
   4,  9,  9,  9,  9,  9
  10,  6,  6,  4,  4, 10
  11, 11, 11, 11, 11, 11
   8, 12, 12, 12, 12, 12
  13, 13, 13, 13, 13, 13
  14, 14, 14, 14, 14, 10
  10,  9,  9,  6,  6, 15

Links

Crossrefs

Cf. A000040, A084558, A099563, A249344.
Cf. also A007623, A275723.

Programs

Formula

A(n,k) = (n / prime(1+A084558(k))^e) * prime(1+A084558(k)-A099563(k))^e, where e = A249344((1+A084558(k)), n), the exponent of the largest power prime(1+A084558(k)) which divides n.
Showing 1-3 of 3 results.

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